Recent research findings on M87 (NGC 4486)

M87 (Messier 87), also known as NGC 4486, is a giant elliptical galaxy, located about 53.5 million light-years away. It is noteworthy for several reasons, including the presence of an unusually large supermassive black hole (SMBH) in its active galactic nucleus, with an estimated mass of about 6.4×10⁹ times the mass of the Sun (M_⊙), two plasma jets that emit strongly at radio frequencies and extend at least 5000 light-years from the SMBH (although only the jet pointed more towards us is readily detectable), and a population of about 15,000 globular clusters.

The total mass of M87 is difficult to estimate, because elliptical galaxies like M87, and unlike spiral galaxies, do not tend to follow the Tully-Fisher relation between intrinsic luminosity and total mass calculated from rotation curves – which therefore includes dark matter. Estimates of the total mass of M87, including dark matter, come in around 6×10¹² M_⊙ within a radius of 150,000 light-years from the center. This compares with about 7×10¹¹ M_⊙ for the Milky Way, but M87 could be more than 10 times as massive.

In other comparisons, the Milky Way has only about 160 globular clusters, and a central black hole (Sagittarius A*) with a mass of about 4.2×10⁶ M_⊙. So M87's central black hole is about 1500 times as massive as the Milky Way's. Pretty impressive difference.



M87 – click for 640×480 image

Besides the recent research listed below, I've written about earlier research on M87 in these articles: Galactic black holes may be more massive than thought, Stellar birth control by supermassive black holes, Black holes in the news.

You might also be interested in some articles from the past year on the general subject of active galaxies: Active galaxies and supermassive black hole jets, Where the action is in black hole jets, Quasars in the very early universe.

Feedback under the microscope: thermodynamic structure and AGN driven shocks in M87 (6/29/10) – arXiv paper

Feedback under the microscope II: heating, gas uplift, and mixing in the nearest cluster core (3/28/10) – arXiv paper

Activity of the SMBH in M87 has a significant effect not only on the host galaxy, but also on the Virgo cluster of galaxies in which M87 is near the center. Energetic outflows of matter from near the black hole force plumes of gas out of the galaxy into the hotter intergalactic medium. The mass transported in this way represents about as much gas as is contained within 12,000 light-years of M87's center. (However, that's only about 2.5% of M87's 500,000 light-year radius.) If it had not been expelled, the gas could have formed hundreds of millions of stars.

The first paper reports on studies using the Chandra X-ray Observatory to measure gas temperatures around M87's center. The findings include detection of 2 distinct shock wave fronts about 46 thousand light-years and 10 thousand light years from the center. This indicates that explosive events occurred about 150 million and 11 million years ago, respectively.

The second paper uses observations from Chandra, XMM-Newton, and optical spectra to distinguish different phases of the hot gas surrounding M87's SMBH.

Refs:
• Galactic 'Super-Volcano' in Action (8/20/10) – Science Daily (press release)
• Galactic Supervolcano Erupts From Black Hole (8/20/10) – Wired.com
• Galactic 'Supervolcano' Seen Erupting With X-Rays (9/6/10) – Space.com

A correlation between central supermassive black holes and the globular cluster systems of early-type galaxies (8/13/10) – arXiv paper

A study of 13 galaxies, including M87, has found a correlation between the size of a galaxy's SMBH and the number of the galaxy's globular clusters. The types of galaxies studied included nine giant ellipticals (like M87), a tight spiral, and 3 galaxies intermediate in type between spiral and elliptical. The smallness of the sample is due to the exclusion of open spiral galaxies and the further limitation to cases where good estimates of the number of globular clusters and mass of the central black hole existed.

The correlation, in which the number of globular clusters is proportional to the black hole mass, is actually stronger than correlations between black hole mass and other galaxy properties previously studied for correlation, such as stellar velocity dispersion (an indicator of total mass), and luminosity of the galaxy's central bulge or whole galaxy (for ellipticals).

In some cases the correlation of black hole mass with total luminosity was especially weak, but better with number of globular clusters. For instance, Fornax A (NGC 1316) is a giant lenticular galaxy with luminosity comparable to that of M87. Yet its central black hole has a mass of 1.5×10⁸ M_⊙, 2.3% that of M87's black hole. It has 1200 globular clusters, 8% of M87's count. Clearly this is not a linear relation. Rather, the study found that the best fit was a power law with M_• ≈ (1.7×10⁵)×N^1.08±0.04, where M_• is black hole mass in units of M_⊙ and N is number of globular clusters. This relation predicts a SMBH mass of 5.5×10⁹ M_⊙ for M87, which is very close, and 3.6×10⁸ M_⊙ for the SMBH mass of NGC 1316, which is high – but the SMBH mass of NGC 1316 is also unusually low in comparison with its luminosity and velocity dispersion.

By contrast, the relation predicts that the Milky Way with a SMBH mass of 4.2×10⁶ M_⊙ should have only about 20 globular clusters, while the actual number is about 160. However, the Milky Way is a loose spiral, not one of the types that was studied, which may account for the much worse correlation. The fit is much better if only globular clusters associated with the central bulge (about 30) are considered.

The obvious question is about why this relation between SMBH mass and number of globular clusters exists. Presumably it has much to do with the typical history of a large galaxy, which is expected to include frequent mergers with other galaxies. The existence of the relationship should provide clues to galactic history, and especially how this may be different for loose spirals like the Milky Way, in comparison with more compact galaxies.

Refs:
• A correlation between central supermassive black holes and the globular cluster systems of early type galaxies (8/11/10) – The Astrophysical Journal
• Supermassive black holes reveal a surprising clue (5/25/10) – Physicsworld.com

A Displaced Supermassive Black Hole in M87 (6/16/10) – arXiv paper

It has generally been assumed that a galaxy's central SMBH is very close to the actual center of mass of the galaxy, because that is (by definition) the gravitational equilibrium point. This central point should be essentially the same as the photometric center of the galaxy, since the galaxy's stars should be distributed symmetrically around the center. Consequently, astronomers have not carefully searched for cases where a SMBH is not very near the galactic center. This lack of extensive investigation is also a result of the fact that the SMBH is often hidden inside a dense cloud of dust, so its exact position is difficult to determine. M87's SMBH (more precisely, the accretion disk around the SMBH), however, is clearly visible, and the research reported in this paper finds it is actually located about 22 light-years from the apparent galactic center.

There are various possible reasons for this much displacement from the center, and not a lot of evidence to identify the most likely reason. Possible reasons include: (1) The SMBH is part of a binary system in which the other member is not detected. (2) The SMBH could have been gravitationally perturbed by a massive object such as a globular cluster. (3) There is a significant asymmetry of the jets. (4) The SMBH has relatively recently merged with another SMBH, subsequent to an earlier merger of another galaxy with M87.

The displacement of the SMBH is in the direction opposite the visible jet, so the last two possibilities are more likely than the others. However, possibility (3) depends on the jet structure having existed at least 100 million years and the density of matter at the center of M87 being low enough to provide insufficient restoring force. Possibility (4) is viable if the SMBH is still oscillating around the center following a galactic merger within the past billion years.

Refs:
• A Displaced Supermassive Black Hole in M87 (6/9/10) – The Astrophysical Journal Letters
• Black Hole Shoved Aside, Along With 'central' Dogma (5/25/10) – Science News
• Black Hole Found in Unexpected Place (5/25/10) – Wired.com
• Supermassive black holes may frequently roam galaxy centers (5/25/10) – Physog.com (press release)
• Bizarre Behavior of Two Giant Black Holes Surprises Scientists (5/25/10) – Space.com
• Galactic Black Holes Can Migrate or Quickly Awaken from Quiescence (5/26/10) – Scientific American




M87 jet

Radio Imaging of the Very-High-Energy γ-Ray Emission Region in the Central Engine of a Radio Galaxy (7/24/09) – Science

Energetic plasma jets, in which matter is accelerated close to the speed of light, combined with intense electromagnetic emissions, especially at radio frequencies, are prominent in about 10% of active galaxies, including M87. However, little has been well established about what processes are responsible for the emissions, or more generally how the jets are powered, accelerated, and focused into narrow beams. Because of the relative proximity of M87 and the fact that the jet we observe is angled from 15° to 25° to our line of sight, M87 is one of the best objects to study in order to learn more about how jets work.

Gamma rays, because of their very high energies (greater than 100 keV per photon), are not continuously produced in active galaxy jets, but are occasionally observed in short bursts lasting only a few days. One such event occurred in M87 in February 2008. At the same time, the intensity of radiation at all other wavelengths increased substantially. Such flares, at lower energies, are not unusual, since the energy output of most jets is somewhat variable in time. The flare persisted for much longer at energies below the gamma-ray band, indicating that the disturbance continued to propagate along the jet even after the gamma-ray flare subsided. However, although we don't know what the cause was, the coincidence in time of the gamma-ray emissions and the beginning of the extended flare makes it very likely that the events had the same source.

This is significant information, because our technology for detecting gamma-ray events has very poor angular resolution (~0.1°), since gamma rays can be detected on the ground only by secondary effects that a gamma ray produces in our upper atmosphere. More than 6 orders of magnitude finer resolution can be achieved at radio frequencies, using very long baseline interferometry. With that technology, it was possible to locate the origin of the disturbance that caused both gamma ray and lower energy flaring to a region within about 100 Schwarzschild radii (R_s) of the SMBH. Since R_s = 2G×M_•/c², R_s for the M87 SMBH is about 1.9×10¹⁰ km, or more than twice the radius of the solar system. So 100R_s is about 70 light-days – which is pretty small compared to the 53.5 million light-year distance to M87.

It's also significant that the gamma-ray event occurred so close to the SMBH, because the cause must be unlike whatever is responsible for the flaring described in the following research.

Refs:
• VLBA locates superenergetic bursts near giant black hole (7/2/09) – Physorg.com (press release)
• Mysterious Light Originates Near A Galaxy's Black Hole (7/2/09) – Space.com
• A Flare for Acceleration (7/24/09) – Science
• High Energy Galactic Particle Accelerator Located (9/14/09) – Science Daily (press release)

Hubble Space Telescope observations of an extraordinary flare in the M87 jet (4/22/09) – arXiv paper

Electromagnetic radiation from SMBH jets is fairly variable in both time and location along the jet. In the case of M87, high-resolution images at various wavelengths have shown the existence of many regions of enhanced emissions within the jet. One of the most prominent of these even has a name: HST-1, so-named because it was discovered by the Hubble Space Telescope. It occupies a stationary position on the jet, about a million Schwarzschild radii from the center, i. e. about 2000 light-years from the SMBH.

HST-1 has been observable for some time, but until February 2000 it was relatively dormant. After that it began to flare more brightly across the electromagnetic spectrum up to X-rays. In 2003 it became more variable, and it reached its greatest brightness in May 2005, when the flux in near ultraviolet was 4 times as great as that of M87's central energy source, the SMBH accretion disk. This represents a brightness increase at that wavelength of a factor of 90. The X-ray flux increased by a factor of 50, and similar, synchronized changes occurred at other wavelengths. The synchronization indicates that one mechanism is responsible for the variability at all wavelengths.

What the actual cause of the disturbance may be is not clear. Because of the great distance of HST-1 from the SMBH, its basic energy source must not be the central accretion disk itself. More likely HST-1 is a result of constriction of magnetic field lines, resulting in further acceleration of the particles making up the jet. Acceleration of charged particles causes radiation by the synchrotron process, and is evidenced by polarization of the emitted photons. Constriction of the jet may be a result of passage through a region of higher density of stars. The increased variability could mean that the jet has encountered a region of higher but varying stellar density. Alternatively, the jet may be passing through a patch of thick gas or dust, with excess radiation produced by the resulting particle collisions.

These results could explain the variability of light from other, more distant active galaxies, at least those which have strong jets, given that it's possible for a small region of the jet far from the SMBH to outshine the central source. However, another source of variability occurs when a jet is viewed at a very low angle to our line of sight, in which case any slight change of direction could cause an apparent change of brightness.

Refs:
• Hubble Space Telescope observations of an extraordinary flare in the M87 jet (3/6/09) – The Astronomical Journal
• Hubble Witnesses Spectacular Flaring in Gas Jet from M87's Black Hole (4/14/09) – Physorg.com (press release)
• Black Hole Creates Spectacular Light Show (4/14/09) – Space.com
• Black hole jet brightens mysteriously (4/15/09) – New Scientist
• Black hole spews out impressive light show (4/20/09) – Cosmos Magazine

Pinwheel of Star Birth

Pinwheel of Star Birth (10/19/10)

This face-on spiral galaxy, called NGC 3982, is striking for its rich tapestry of star birth, along with its winding arms. The arms are lined with pink star-forming regions of glowing hydrogen, newborn blue star clusters, and obscuring dust lanes that provide the raw material for future generations of stars. The bright nucleus is home to an older population of stars, which grow ever more densely packed toward the center.

NGC 3982 is located about 68 million light-years away in the constellation Ursa Major. The galaxy spans about 30,000 light-years, one-third of the size of our Milky Way galaxy.

NGC 3982 – click for 984×1000 image

More: here

Roots of unity and cyclotomic fields

In preparation for many good things that are to come, we need to have a talk about another important class of field extensions of ℚ – the cyclotomic extensions. (Check here for a list of previous articles on algebraic number theory.)

A cyclotomic field in general is a field that is an extension of some base field formed by adjoining all the roots of the polynomial f(x) = xⁿ-1=0 for some specific positive n∈ℤ to the base field. Usually, though not always, this will mean roots that lie in some large field in which f(x) splits completely and that contains ℚ as the base field, such as ℂ, the complex numbers. f(x) is known as the n^th cyclotomic polynomial. Mostly the same theory applies if the base field is a finite algebraic extension of ℚ, but we'll use ℚ as the base field for simplicity.

Since f(1)=0, x-1 is one factor of f(x), and f(x)/(x-1) = x^n-1 + … + x + 1 ∈ ℤ[x], with all coefficients equal to 1. If n is even, -1 is also a root of f(x). However, all other roots of f(x) in ℂ are complex numbers that are not in ℝ. Some of these roots, known as the "primitive" n^th roots of unity – denoted by ζ_n (or just ζ if the context is clear) – have the property that all other roots are a power ζ^k for some integer k, 1≤k<n. So the smallest subfield of ℂ that contains ℚ and all roots of f(x) is ℚ(&zeta), known as the n^th cyclotomic field.

It is possible to express all the roots of f(x) in the form e^2πi/n, where e^z is the complex-valued exponential function, which can be defined in various ways. The most straightforward way is in terms of an infinite series, e^z = Σ_0≤n<∞zⁿ/n!. The exponential function e^z can also be defined as the solution of the differential equation dF(z)/dz = F(z) with initial value F(1)=e, the base of the natural logarithms. So there is the rather unusual circumstance that the roots of an algebraic equation can be expressed as special values of a transcendental function. Mathematicians long hoped that other important examples like this could be found (a problem sometimes referred to as "Kronecker's Jugendtraum", a special case of Hilbert's twelfth problem), but that hope has mostly not been fulfilled.

The most well-known nontrivial root of unity is the fourth root, i=&radic(-1), which satisfies x⁴-1 = (x²+1)(x²-1) = 0.

All complex roots of unity have absolute value 1, i. e. |ζ|=1, since |ζ| is a positive real number such that |ζ|ⁿ=1. The set of all complex numbers with |z|=1 is simply the unit circle in the complex plane, since if z=x+iy, then |z|² = x²+y² = 1. (Note that the linguistic root of words like "circle", "cyclic", and "cyclotomic" is the Greek κύκλος (kuklos).) Since e^iθ = sin(θ) + i⋅cos(θ) for any θ, with θ=2πk/n the real and imaginary parts of a general n^th root of unity ζ=e^2πi⋅k/n are just Re(ζ)=sin(2πk/n) and Im(ζ)=i⋅cos(2πk/n).

There are many reasons why cyclotomic fields are important, and we'll eventually discuss a number of them. One simple reason is that roots of algebraic equations can sometimes be expressed in terms of real-valued roots (such as cube roots, d^1/3 for some d), and roots of unity. See, for example, this article, where we discussed the Galois group of the splitting field of f(x)=x³-2.

The set of all complex n^th roots of unity forms a group under multiplication, denoted by μ_n. This group is cyclic, of order n, generated by any primitive n^th roots of unity. (Any finite subgroup of the multiplicative group of a field is cyclic.) As such, it is isomorphic to the additive group ℤ_n = ℤ/nℤ, the group of integers modulo n. Because of this, many of the group properties of μ_n are just restatements of number theoretic properties of ℤ_n. For instance, each element of order n in μ_n is a generator of the whole group – one of the primitive n^th roots of unity. Since μ_n⊆ℚ(ζ_n), adjoining all of μ_n gives the same extension ℚ(ζ_n) = ℚ(μ_n).

Now, ℤ/nℤ is a ring, and its elements that are not divisors of zero are invertible, i. e they are units of the ring. They form a group under ring multiplication, which in this case is written as (ℤ/nℤ)^× (sometimes U_n for short). An integer m is invertible in ℤ/nℤ if and only if it is prime to n, i. e. (m,n)=1 (because of the Euclidean algorithm). The number of such distinct integers modulo n is a function of n, written φ(n). This number is important enough to have its own symbol, because it was studied by Euler as fundamental to the arithmetic of ℤ/nℤ. Thus φ(n) is also the order of the group (ℤ/nℤ)^×.

Let ζ=e^(2πi)m/n, for 0≤m<n, be an element of μ_n. The correspondence m↔e^(2πi)m/n establishes a group isomorphism between the additive cyclic group ℤ/nℤ and the multiplicative group μ_n. Modulo n, m generates ℤ/nℤ additively if and only if (m,n)=1, which is if and only if the corresponding ζ generates μ_n. So the number of generators of μ_n – which is the number of primitive n^th roots of unity – is the same as the order of (ℤ/nℤ)^×, i. e. φ(n).

One has to be careful, because the multiplicative structure of μ_n parallels the additive structure of ℤ/nℤ, not the multiplicative structure of (ℤ/nℤ)^×. (Because if ζ^M and ζ^N are typical elements of μ_n then ζ^M×ζ^N=ζ^M+N.) Hence even though there are φ(n) generators of μ_n, these generators do not form a group by themselves (a product of generators isn't in general a generator), so the set of them isn't isomorphic to (ℤ/nℤ)^×, even though the latter also has φ(n) elements. Give this a little thought if it seems confusing.

Moreover, the group (ℤ/nℤ)^× is not necessarily cyclic. It is cyclic if n is 1, 2, 4, p^e, or 2p^e for odd prime p, but not otherwise. Confusingly, if the group does happen to be cyclic then integers modulo n that generate the whole group are called "primitive roots" for the integer n. If (ℤ/nℤ)^× happens to be cyclic, then only those m∈(ℤ/nℤ)^× having order φ(n) are "primitive roots" that generate the group, while all m∈(ℤ/nℤ)^× have the property that if ζ∈μ_n has order n and generates the latter group, then so does ζ^m, as we showed above. Got that straight, now? This needs to be understood when working in detail with roots of unity.

Another reason for the importance of cyclotomic fields is that the Galois group of the extension [ℚ(ζ_n):ℚ] is especially easy to describe. Indeed, it is isomorphic to the group of order φ(n) we've just discussed: (ℤ/nℤ)^×. There's a little work in proving this isomorphism, but let's first note what it implies. Let G=G(ℚ(ζ)/ℚ) be the Galois group. It is an abelian group of order φ(n) since it's isomorphic to (ℤ/nℤ)^×. Further, any subgroup of G′ of G is abelian and by Galois theory determines an abelian extension (i. e., an extension that is Galois with an abelian Galois group) of ℚ as the fixed field of G′. Conversely, it can be shown (not easily) that every abelian extension of ℚ is contained in some cyclotomic field. (This is the Kronecker-Weber theorem.)

Half of the proof of the isomorphism is easy. Pick one generator ζ of μ_n, i. e. a primitive n^th root of unity. We'll see that it doesn't matter which of the φ(n) possibilities we use. Suppose σ∈G is an automorphism in the Galois group. Since σ is an automorphism and ζ generates the field extension, all we need to know is how σ acts on ζ. Since σ is an automorphism, σ(ζ) has the same order as ζ, so it's also a primitive n^th root of unity. Therefore &sigma(ζ) = ζ^m for some m, 1≤m<n. As we saw above, m is uniquely determined and has to be a unit of ℤ/nℤ, with (m,n)=1, in order for ζ^m to be, like ζ, a generator of the cyclic multiplicative group μ_n. Hence m∈(ℤ/nℤ)^×. Call this map from G to (ℤ/nℤ)^× j, so that σ(ζ)=ζ^j(σ). To see that it's a group homomorphism, suppose σ₁,σ₂∈G, with j(σ₁)=r, j(σ₂)=s. Then σ₂(σ₁(ζ)) = σ₂(ζ^r) = (ζ^s)^r = ζ^sr, hence j(σ₂σ₁) = j(σ₂)j(σ₁). j is clearly injective since j(σ)=1 means σ(ζ)=ζ, so σ is the identity element of G. Finally, to see that j doesn't depend on the choice of primitive n^th root of unity, suppose ζ^m with m∈(ℤ/nℤ)^× is another one. Then σ(ζ^m) = σ(ζ)^m = (ζ^j(σ))^m = (ζ^m)^j(σ).

Thus G is isomorphic to a subgroup of (ℤ/nℤ)^×. That's enough to show G is abelian, so the extension ℚ(ζ)/ℚ is abelian. To complete the proof of an isomorphism G≅(ℤ/nℤ)^× we would need to show that the injective homomorphism j is also surjective, i. e. every m∈(ℤ/nℤ)^× determines some σ∈G such that m=j(σ). We can certainly define a function from ℚ(ζ) to ℚ(ζ) by σ(ζ)=ζ^m for a generator ζ of the field ℚ(ζ). One might naively think that's enough, but the problem is that one has to show that σ is a field automorphism of ℚ(ζ).

The map σ defined that way certainly permutes the n^th roots of unity in μ_n, the roots of the polynomial f(x)=xⁿ-1. However, not all permutations of elements of μ_n, of which there are n!, yield automorphisms of ℚ(ζ). The problem here is that if z(x) is the minimal polynomial of some ζ, i. e. the irreducible polynomial of smallest degree in ℤ[x] such that z(ζ)=0, then by Galois theory the order |G| of the Galois group G is the degree of the field extension, which is the degree of z(x). Since G is isomorphic to a subgroup of the group (ℤ/nℤ)^×, and the latter has order φ(n), all we know is that |G| divides φ(n). It could be that other primitive n^th roots of unity have minimal polynomials in ℤ[x] that are not the same as z(x), though they have the same degree |G|. For σ to be an automorphism, σ(ζ) needs to have the same minimal polynomial as ζ, and we don't know that immediately from the relation σ(ζ)=ζ^m.

We will defer discussion of the rest of the proof that G(ℚ(μ_n)/ℚ)≅(ℤ/nℤ)^× for the next installment, since some new and important concepts will be introduced.

Splitting of prime ideals in algebraic number fields

Our series of articles on algebraic number theory is back again. Maybe this time it won't be so sporadic. Stranger things have happened. The previous installment, of which this is a direct continuation, is here. All previous installments are listed here.

When we left off, we were talking about how to determine the way a prime ideal factors in the ring of integers of a quadratic extension of ℚ. Such a field is of the form ℚ(√d) for some square-free d∈ℤ. We were using very simple elementary reasoning with congruences, and we found a fairly simple rule, namely:

If p∈ℤ is an odd prime (i. e., not 2), and K=ℚ(√d) is a quadratic extension of ℚ (where d is not divisible by a square) then

1. p splits completely in K if and only if p∤d and d is a square modulo p.
2. p is prime (i. e. inert) in K if and only if d is not a square modulo p.
3. p is ramified in K if and only if p|d.

The prime 2 behaves a little more weirdly, but the result is that 2 ramifies if and only if d≡2 or 3 (mod 4); 2 is inert if and only if d≡5 (mod 8); 2 splits if and only if d≡1 (mod 8).

One limitation was that our simple reasoning made it necessary to assume that O_K, the ring of integers of K, was a PID (principal ideal domain).

Let's review what we were trying to do. We were investigating the factorization of a prime ideal (p)=pO_ℚ(√d) in O_ℚ(√d). If O_ℚ(√d) is a PID, then there is a simple approach to investigate how p splits. If p splits then (p)=P₁⋅P₂, where P_i=(α_i), i=1,2. Any quadratic extension is Galois, and the Galois group permutes the prime ideal factors of (p). The factors are conjugate, so if α₁=a+b√d we can assume α₂=α₁*=a-b√d. Hence (p)=(α₁)⋅(α₁*)= (α₁α₁*)= (a²-db²).

Taking norms (to eliminate possible units ε∈O_ℚ(√d)) reduces the problem to a Diophantine equation of the form ±p=a²-db². With the problem thus reduced, a necessary condition for (p) to split (or ramify) is that the equation can be solved for a,b∈ℤ. A sufficient condition to show that (p) is inert, i. e. doesn't split or ramify, is to show that the equation can't be solved.

Let's look at how that might work. For example, let d=3. Looking at the equations modulo 3, we have ±p≡a² (mod 3). That is, either p or -p is a square modulo 3. Say p=5. The only nonzero square mod 3 is 1, and 5≢1 (mod 3). However -5≡1 (mod 3), so could we have -5=a²-3b²? Suppose there were some a,b∈ℤ such that -5=a²-3b². Then instead of looking at the equation modulo 3, we could look at it modulo 5, and find that then a²≡3b² (mod 5). If 5 divides either a or b, it divides both, and so 25 divides a²-3b², which is impossible since 25∤5. Therefore 5∤b. ℤ/(5) is a field, so b must have an inverse c such that cb≡1 (mod 5). Therefore, (ac)² ≡ 3(bc)² ≡ 3 (mod 5), and so 3 is a square mod 5. But that can't be, since only 1 and 4 are squares modulo 5. The contradiction implies -5=a²-3b² has no solution for a,b∈Z.

All that does show 5 doesn't split or ramify in ℚ(√3), hence it must be intert, but this approach is messy and still requires knowing that the integers of ℚ(√3) form a PID. We need to find a better way. Fortunately, there is one. But first let's observe that this elementary discussion shows there is a fairly complicated interrelationship among:

1. Factorization of (prime) ideals in extension fields,
2. Whether a given ring of integers is a PID,
3. Whether an integer prime can be represented as the norm of an integer in an extension field,
4. Whether an integer can be represented by an expression of the form a²+db² for a,b∈Z (in the case of quadratic extensions),
5. Whether, for primes p,q∈Z, p is a square modulo q and/or q is a square modulo p.

The problem of representing an integer by an expression like a²+db² is a question of solving a Diophantine equation, and more specifically is of the type known as representing a number by the value of a quadratic form. This question was studied extensively by Gauss, who proved a remarkable and very important result, known as the law of quadratic reciprocity, which relates p being a square modulo q to q being a square modulo p, for primes p,q.

We will take up quadratic reciprocity soon (and eventually much more general "reciprocity laws"), but right now, let's attack head on the issue of determining how a prime of a base field splits in the ring of integers of an extension field. We will use abstract algebra instead of simple arithmetic to deal with this question. For simplicity, we'll assume here that the base field is ℚ, even though many results can be stated, and are often valid, for more arbitrary base fields.

Chinese Remainder Theorem

The first piece of abstract algebra we'll need is the Chinese Remainder Theorem (CRT). Although it's been known since antiquity to hold for the ring ℤ, generalizations are actually true for any commutative ring.

Let R be a commutative ring, and suppose you have a collection of ideals I_j, for j in some index set, j∈J. Suppose that the ideals are relatively prime in pairs. In general that means that I_i+I_j=R if i≠j, and further, the product of ideals, I_i⋅I_j, is I_i∩I_j when i≠j. If R is Dedekind, then each ideal has a unique factorization into prime ideals, and they are relatively prime if I_i and I_j have no prime ideal factors in common when i≠j. Let I be the product of all I_j for j∈J, which is also the intersection of all I_j for j∈J, since the ideals are coprime in pairs.

The direct product of rings R_i for 1≤i≤k is defined to be the set of all ordered k-tuples (r₁, ... ,r_k), for r_i∈R_i, with ring structure given by element-wise addition and multiplication. The direct product is written as R₁×...×R_k, or &Pi_1≤i≤kR_i.

Given all that, the CRT says the quotient ring R/I is isomorphic to the direct product of quotient rings &Pi_1≤i≤k(R/I_i) via the ring homomorphism f(x)=(x+I₁, ... ,x+I_k) for all x∈R.

The CRT is very straightforward, since f is obviously a surjective ring homomorphism, and the kernel is I, since it's the intersection of all I_i. (It's straightforward, at least, if you're used to concepts like "surjective" and "kernel".)

Now we'll apply the CRT in two different situations. First let R be the ring of integers O_K of a finite extension K/ℚ, and I_i=P_i, 1≤i≤g, be the set of all distinct prime ideals of O_K that divide (p)=pO_K for some prime p∈ℤ. Then (p)=P₁^e₁ ⋅⋅⋅ P_g^e_g, where e_i are the ramification indices of each prime factor of (p). An application of CRT then shows that O_K/(p) ≅ Π_1≤i≤g(O_K/P_i^e_i). Recall that for each i, O_K/P_i is isomorphic to the finite field F_qi, where q_i=p^f_i for some f_i, known as the degree of inertia of P_i. (This field is the extension of degree f_i of F_p=ℤ/pℤ.) Further, Σ_1≤i≤ge_if_i=[K:ℚ], the degree of the extension. Check here if you need to review these facts. Specifying how (p) splits in O_K amounts to determination of the P_i and the numbers e_i, f_i, and g.

The second situation where we apply CRT involves the ring of polynomials in one variable over the finite field F_p=ℤ/pℤ, denoted by F_p[x]. Let f(x) be a monic irreducible polynomial with integer coefficients, i. e. an element of ℤ[x]. Let f(x) be f(x) with all coefficients reduced modulo p, an element of F_p[x]. f(x) will not, in general, be irreducible in F_p[x], so it will be a product of powers of irreducible factors: Π_1≤i≤g(f_i(x)^e_i), where f_i(x)∈F_p[x]. Each quotient ring F_p[x]/(f_i(x)) is a finite field that is an extension of F_p of some degree f_i. In general, e_i, f_i, and g will be different, of course, from the same numbers in the preceding paragraph. But the CRT gives us an isomorphism F_p[x]/(f(x)) ≅ &Pi_1≤i≤g(F_p[x]/(f_i(x)^e_i)).

Now, here's the good news. For many field extensions K/ℚ, there exists an appropriate choice of f(x)∈ℤ[x] such that for most primes (depending on K and f(x)), the numbers e_i, f_i, and g will be the same for both applications of the CRT. Consequently, we will have O_K/(p) ≅ F_p[x]/(f(x)), because for corresponding factors of the direct product of rings, O_K/P_i^e_i ≅ F_p[x]/(f_i(x)^e_i). As it happens, most primes don't ramify for given choices of K and f(x), so that things are even simpler, since all e_i=1, and all factors of the direct products are fields.

We can't go into all of the details now as to how to choose f(x) and what the limitations on this result are. However, here are the basics. Any finite algebraic extension of ℚ (and indeed of any base field that is a finite algebraic extension of ℚ) can be generated by a single algebraic number θ: K=ℚ(θ), called a "primitive element". In fact, &theta can be chosen to be an integer of K. Then the ring of integers of K, O_K, is a finitely generated module over ℤ. (A module is like a vector space, except that all coefficients belong to a ring rather than a field.) The number of generators is the index [O_K:ℤ[θ]]. (ℤ[θ] is just all polynomials in θ with coefficients in ℤ.) If p∈ℤ is any prime that does not divide [O_K:ℤ[θ]], then the result of the preceding paragraph holds. If for some p and some choice of θ p does divide the index, then there may be another choice of θ for which p doesn't divide the index. Unfortunately, there are some fields (even of degree 3 over ℚ) where this isn't possible for some choices of p.

The situation is especially nice in the case of quadratic fields, K=ℚ(√d), square-free d∈ℤ. If d≢1 (mod 4); we can take θ=√d and f(x)=x²-d, since O_K=ℤ[√d]. If d≡1 (mod 4), then the index [O_K:ℤ[√d]]=2, and there's a possible problem only for p=2. However, we still have O_K/(p) ≅ F_p[x]/(x²-d) for all p≠2. From that it's obvious that, except for p=2, (p) ramifies if p|d, (p) splits if d is a square modulo p, or else (p) is inert. That is exactly the conclusion we began with at the beginning of this article, on the basis of elementary considerations. Only now we need not assume that O_K is a PID.

There are four important lessons to take away from this discussion.

First, there is a very close relationship between the arithmetic of algebraic number fields and the arithmetic of polynomials over a finite field. Not only do we have the isomorphism discussed above, but it turns out that a number of similar powerful theorems are true for both algebraic number fields and the field of quotients of polynomial rings over a finite field.

Second, a lot of the arithmetic of algebraic number fields can be analyzed in terms of what happens "locally" with the prime ideals of the ring of integers of the field.

Third, many of the results of algebraic number theory are fairly simple if the rings of integers are PIDs (or, equivalently, have unique factorization). Such results often remain true when the rings aren't PIDs, though they can be a lot harder to prove. Often the path to proving such results involves considering the degree to which a given ring of integers departs from being a PID.

Fourth, and perhaps most importantly, abstract algebra is a very powerful tool for understanding algebraic number fields – and it is much easier to work with and understand than trying to use "elementary" methods with explicit calculations involving polynomials and their roots.

We will see these lessons validated time and again as we get deeper into the subject.

So where do we go from here? There are a lot of directions we could take, so we'll probably jump around among a variety of topics.

Forearms.

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Monday Randoms.

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Back.

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Monday Randoms.

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The sweetest thing.

Source: twitterpic

Friday Randoms.

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I really like the middle one. I'll try my best to respond to formspring and comments over the weekend!

Words.

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Arms

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Arm.

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Selected readings 11/28/10

Interesting reading and news items.

Please leave some comments that indicate which articles you find most interesting or that identify topics you would like to read about, and I will try to include more articles of a similar nature in the future

These items are also bookmarked at my Diigo account.

Bad seeds, bad science, and fairly black cats?

Geneticists have failed to remind the public what the word “genetic” actually means. Heritability implies that gene and environment work, or might be persuaded to work, together. Why, after all, are taxpayers spending money on the double helix if there is no hope of an environmental intervention—a drug, a change in lifestyle, or cancer surgery after the early diagnosis of a somatic mutation—to help those at risk from what they inherit? Everyone in the trade knows this although they fail to mention it except to their first-year undergraduate classes. Transcripts of their lectures should be sent out with every press release. [The Lancet, 10/23/10]

Cancer’s little helpers

No one would have predicted a decade ago that these microRNAs, as the hairpins are called, were involved in cancer, because no one even knew that they existed in people. Mere snippets of RNA — DNA’s underappreciated cousin — these micromolecules are about 22 chemical letters long. But their size belies their power. [Science News, 8/28/10]

Hogan’s holometer: Testing the hypothesis of a holographic universe

In 2008, Fermilab particle astrophysicist Craig Hogan made waves with a mind-boggling proposition: The 3D universe in which we appear to live is no more than a hologram. Now he is building the most precise clock of all time to directly measure whether our reality is an illusion. [Symmetry Breaking, 10/20/10]

The Brain That Changed Everything

When a surgeon cut into Henry Molaison's skull to treat him for epilepsy, he inadvertently created the most important brain-research subject of our time — a man who could no longer remember, who taught us everything we know about memory. Six decades later, another daring researcher is cutting into Henry's brain. Another revolution in brain science is about to begin. [Esquire, 10/25/10]

How Big is the Unobservable Universe?

Based on what we currently think about inflation, this means that the Universe is at least 10^(1030) times the size of our observable Universe! And good luck living long enough to even write that number down. ... All that we know, see, and observe is just one tiny region that slid down that hill fast enough to end inflation, but most of it just keeps on inflating forever and ever. [Starts with a Bang!, 10/27/10]

Revealing the galaxy’s dark side

“In our paper, we discussed a number of astrophysical possibilities for the origin of the signal, including a population of pulsars, cosmic ray interactions and emission from our galaxy's supermassive black hole,” notes Hooper. “And in the end, no combination of any astrophysical sources could give us the signal we’re seeing,” he adds. “Eventually we just got fed up and concluded there doesn’t seem to be a way to explain the signal except for one thing — we tried dark matter and it fit beautifully without any special bells or whistles.” [Science News, 11/20/10]

When Muons Collide

A new type of particle collider known as a muon collider considered a wild idea a decade ago is winning over skeptics as scientists find solutions to the machine's many technological challenges. [Symmetry, 10/1/10]

We all need (a little bit of) sex

Sex costs amazing amounts of time and energy. Just take birds of paradise touting their tails, stags jousting with their antlers or singles spending their weekends in loud and sweaty bars. Is sex really worth all the effort that we, sexual species, collectively put into it? [Scientific American, 11/2/10]

Glia: The new frontier in brain science

Glia, in contrast to neurons, are brain cells that do not generate electrical impulses, and there are a lot of them—85 percent of the cells in the brain. Yet, these cells have been largely neglected for 100 years. I call this new frontier of neuroscience "The Other Brain," because we are only now beginning to explore it. The new findings are expanding our concept of information processing in the brain. They are leading rapidly to new treatments for diseases ranging from spinal cord injury to brain cancer to chronic pain, and Alzheimer's disease. [Scientific American, 11/4/10]

Extra neutrino flavor could be bitter end to Standard Model

What seems to have caught everyone's attention is the suggestion that this might be evidence of what are called sterile neutrinos. Although regular neutrinos barely interact with matter, sterile neutrinos can only interact via gravity, which (if they exist) is what has allowed them to escape our detection to date. Since they'd also be heavier than the regular neutrinos, they would make good dark matter candidates. [Nobel Intent, 11/2/10]

The Neanderthal Romeo and Human Juliet hypothesis

Scientists have had trouble reconciling data from analyses of human mitochondrial DNA and the male Y chromosome. Analyses of human mitochondrial DNA indicate that we all share a common female ancestor 170,000 years ago. Analyses of the Y chromosome indicate that we share a common male ancestor 59,000 years ago. How can we account for the idea that our common grandmother is 111,000 years older than our common grandfather? [Neuroanthropology, 10/26/10]

An idle brain may be the self's workshop

As neuroscientists study the idle brain, some believe they are exploring a central mystery in human psychology: where and how our concept of "self" is created, maintained, altered and renewed. After all, though our minds may wander when in this mode, they rarely wander far from ourselves, as Mrazek's mealtime introspection makes plain. [Los Angeles Times, 8/30/10]

Determining 500th Alien Planet Will Be a Tricky Task

At NASA's last count, astronomers had confirmed the discovery of 494 planets around alien suns. There are signs of dozens more, if not hundreds, but it will take time to weed out which of the detections are actual worlds and which are merely false alarms. [Space.com, 11/11/10]

Tracking Viruses Back in Time

How long have viruses been around? No one knows. Scientists at Portland State University have begun taking the first steps toward answering this question. [Astrobiology, 9/6/10]

Can a 1960s Approach Unify Gravity with the Rest of Physics?

In July mathematicians and physicists met at the Banff International Research Station in Alberta, Canada, to discuss a return to the golden age of particle physics. They were harking back to the 1960s, when physicist Murray Gell-Mann realized that elementary particles could be grouped according to their masses, charges and other properties, falling into patterns that matched complex symmetrical mathematical structures known as Lie groups. [Scientific American, 9/7/10]

Neuroscience: Settling the great glia debate

The consequences of this 'gliotransmission' could be profound. The human brain contains roughly equal numbers of glia and neurons (about 85 billion of each), and any given astrocyte can make as many as 30,000 connections with cells around it. If glia are involved in signalling, processing in the brain turns out to be an order of magnitude more complex than previously expected, says Andrea Volterra, who studies astrocytes at the University of Lausanne in Switzerland. Neuroscientists, who have long focused on the neuron, he says, would have to revise everything. [Nature News, 11/10/10]

This Is Your Brain on Metaphors

Symbols, metaphors, analogies, parables, synecdoche, figures of speech: we understand them. We understand that a captain wants more than just hands when he orders all of them on deck. We understand that Kafka’s “Metamorphosis” isn’t really about a cockroach. If we are of a certain theological ilk, we see bread and wine intertwined with body and blood. We grasp that the right piece of cloth can represent a nation and its values, and that setting fire to such a flag is a highly charged act. [New York Times, 11/14/10]

Tree or ring: the origin of complex cells

All complex life belongs to a single group called the eukaryotes, whose members, from humans to amoebas, share a common ancestry. Their cells are distinguished by having several internal compartments, including the nucleus, which shelters their precious DNA, and the mitochondria, which provide them with power. [Not Exactly Rocket Science, 9/12/10]

I am virus – animal genomes contain more fossil viruses than ever expected

Your closest fossils are inside you, scattered throughout your genome. They are the remains of ancient viruses, which shoved their genes among those of our ancestors. There they remained, turning into genetic fossils that still lurk in our genomes to this day. [Not Exactly Rocket Science, 11/18/10]

Effective Field Theory

"Effective field theory" is a technical term within quantum field theory, but it is associated with a more informal notion of extremely wide applicability. Namely: if we imagine dividing the world into "what happens at very short, microscopic distances" and "what happens at longer, macroscopic distances," then it is possible to consistently describe the macroscopic world without referring to (or even understanding) the microscopic world. [Cosmic Variance, 11/25/10]

Meet a superpartner at the LHC

Of the many ideas for new physics that can be tested at the Large Hadron Collider (LHC), supersymmetry is one of the most promising. The theory proposes that each fundamental fermion particle has a heavier bosonic superpartner (and vice versa for each fundamental boson) and by doing so, offers an extension of the standard model of particle physics that fixes many of its problems. None of the known particles appear to be superpartners, however, which leads to the daunting conclusion that if supersymmetry is correct, there are more than twice as many fundamental particles as we thought, but we have only been left with the lightest partners; that is, supersymmetry is broken. [Physics, 11/22/10]

Mafia Wars

An increasing amount of data is showing that the cellular battle between pathogens and hosts needs much more than a simple military metaphor to describe it—think undercover infiltration, front organizations, and forced suicide. [The Scientist, 6/1/10]

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Finger Tattoo

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Ribs.

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Cole Mohr

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Friday Randoms.

Chests.

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Sunday Randoms.

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Wednesday Randoms

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Disturbing climate change headlines

Yesterday Tom Yulsman at CEJournal came across a story in Fog City Journal that led to a brief post, on which I commented there.

The topic is the fraught question of what's the best way for scientists to respond to global warming Know-Nothingism. My first comment was followed by a response from Tom, and I've responded with a longer note that seems worth sharing here. It turns out that there is a great deal that needs to be said.

What follows is my second response, more or less verbatim.

Tom, I've read the Revkin article and the Feinberg/Willer paper. [See the press release for quick summary.] Thanks for the references. However, I don't find them very persuasive. Apologies in advance for the length of this note.

The Feinberg/Willer paper is based on the social psychology circle of ideas known as "Just World Theory" (JWT). Curiously, the book of the "founder" of JWT, Melvin Lerner, is entitled The Belief in a Just World: A Fundamental Delusion. Unfortunately, I don't have ready access to that volume, but I note that there is no question mark in the title, so I don't know whether Lerner himself actually regarded the underlying "just world" belief as a delusion.

Although the underlying belief that JWT deals with seems philosophically controversial (at best), JWT itself simply asserts that "many people" have this belief, and that certain consequences follow. One thing that concerns me is whether substantial evidence has been developed that quantifies how many people hold the underlying belief in the world's justness. At most it seems like just one dimension in a multidimensional space of belief systems.

It's clear enough that many people have religious beliefs that are incompatible with the idea that a "just" deity would allow the kind of climate developments that science predicts, and so such people deny the science. But that's a pretty broad feature of religion in general – it denies many kinds of science that clash with religion. So what's science supposed to do – give up and say, "Oops. we aren't really predicting what the evidence strongly indicates"?

The Feinberg/Willer paper argues that certain sorts of positive messages increase subjects' acceptance of the ideas (1) that the scientific evidence for global warming is good and (2) that science can find solutions to the problem. In other words, these messages are pro-science in a feel-good, non-threatening way. So of course it's not too surprising that the subjects who heard these messages exhibited greater acceptance of scientific conclusions. This is basic marketing theory.

One problem is that the part of the message that says science can find a "solution" to the problem is likely to be false. It's probable that there is no largely scientific solution. Mitigation of climate change is probably much more of an economic and political issue, because significant behavioral change and economic adjustment are likely to be necessary. Of course, this assertion is also open to debate.

I think that the best science has actually discovered a lot that suggests the threat of climate change is even more dire than some cautious observers assume. There is, for example, this: summary of ten rather disturbing types of climate threat reported in the past year.

You [Tom] wrote, "30 years of unrelenting fear appeals on climate change have gotten us, well, where? I would argue pretty much nowhere. If ever there was a prima facie case that fear appeals on climate change don’t work, this is it."

I'm afraid that by the very same sort of argument, 30 years of attempts to patiently and rationally educate the public on the science of climate change have also failed.

The real problem is that what's actually true is that different approaches work best with different types of people, depending on their undelying personality types and value systems. For example see Skeptics discount science by casting doubts on scientist expertise or the paper it discusses – Cultural cognition of scientific consensus.

One of the individuals that Revkin quotes in his article, Dan Kahan at Yale [and a founder of the Yale Cultural Cognition Project], states the problem quite well:

I think it [Feinberg/Willer] is good research, and maybe captures something that is going on in the real world debate. But it doesn’t capture what’s most important: the source of individual differences. People disagree about climate change; it is one of a cluster of science & policy issues that polarize citizens along cultural/political lines. "Just world" theory posits a general psychological mechanism that affects everyone. Necessarily, then, it can’t explain why one and the same set of informational influences (e.g., stories reporting "scientific consensus" on climate change) provoke different reactions in identifiable subcommunities. The theory that we need is one that identifies what the identifying characteristics of these communities are and how they are implicated in cognition of risk. No theory that focuses of [sic] generic or population-wide aspects of the psychology of risk perception (so-called "main effects") can do that.

In other words, a lot more needs to be done to steer public attitudes in the right direction. It is not a matter of simply finding the most comforting feel-good way to "frame" the issue, if that just entails obscuring the hard scientific facts. That is a vain hope.

I don't have a solution of the problem, but I think a solution should include a careful evidence-based appraisal of the kinds of messages that work best with different groups, combined with a plan for how to deliver the messages through different channels appropriate for different groups.

It's a lot like any other tough political campaign. Sometimes "negative" campaigning works very well, sometimes it doesn't.

I can see what's going on here. There are obviously efforts being made by a broad range of social scientists, communication experts, and journalists to shape an effective messaging strategy. For example: ClimateEngage.org. This is probably good. What is not clear is whether the people most involved will be able to identify a near-optimal strategy.

Just to name names, Matthew Nisbet [also here, here] (whom Revkin also quotes) is one with whom I find a lot to disagree – such as the whole "post-partisan" shtick. The elephant in the room is that most opponents of the necessity of acting on climate change – to say nothing of those who deny it even exists and/or is anthropogenic – have no intentions of operating in a reasonable and responsible "post-partisan" fashion.

There really is a war going on here. Climate scientists who don't face up to this reality are going to get the crap beat out of them. Just ask Phil Jones or Michael Mann [more here], for example. Much like Lt. Colonel George Custer at the Little Big Horn.

Another Favourite.

source: Pernille F