Format: Plan for a 50 minute oral exam.
Gauss Sums and Jacobi Sums (e.g. Chapters 6, 7 Ireland and Rosen)
Finiteness of the Ideal Class Group
Dirichlet’s Unit Theorem
Prime Number Theorem
Dirichlet L-functions (analytic continuation and functional equation)
Dirichlet’s Theorem on Primes in Arithmetic Progressions
Ray Class Groups
Hilbert Class Fields
Dirichlet’s unit theorem
Dirichlet’s theorem on primes in arithmetic progressions
Dirichlet’s class number formula
Prime number theorem
Cebotarev density theorem
Ray class group
Hilbert class fields
Class field theory inequalities
Factorization of ideals
Splitting of primes
Norm of ideals
The discriminant of is , with the plus sign holding if . (Use Vandermonde determinant). By definition, the square root of the discriminant is in the field, so .
Now iff splits completely in (the unique quadratic subfield), and must be , since it is the unique subfield with degree 2 over .
We also know that splits completely in a quadratic field when is a square mod (or for , when ).
Putting this together, we get that .
In the case where , we get that
Generalizes to Artin reciprocity.
For a Dirichlet character modulo , it’s a sum over th roots of unity that has absolute value . So it gives you away to construct a quadratic element in a cyclotomic field.
If , then
These are commonly used to prove quadratic reciprocity.
If is primitive, .
Factors for non-trivial as
Analogous to Beta functions factoring as Gamma functions (which I think is pretty much the prime at infinity case of all this).
We use a similar geometric approach as with Minkowski’s bound, but we take a log map of the lattice space generated by the embeddings so we can look at multiplicative structure. The kernel of this map is the roots of unity contained in the field.
Put a norm on the log space corresponding to the regular norm. The dimension is at most , since units have norm 1, and 1 maps to 0.
Carefully construct a compact, convex, centrally symmetric set satisfying certain inequalities, and with volume equal the volume of the fundamental parallelopiped.
Using Minkowski’s convex body theorem, we can show is guaranteed to contain a lattice point.
We can adjust the inequalities to get lattice points with positive values for all but one dimension. Since the norm of elements of is bounded, by repeatedly applying this to a lattice point, we eventually get points with the same norm, which are different by a unit with certain properties. Doing this for each dimension, we get a suitable set of units.
These units are negative along all dimensions except for one on which they are positive. Putting them in a matrix, we can show that they have rank , which proves our theorem.
Note the similarity of the proof of this theorem and the proof of the finiteness of the ideal class group. These theorems can indeed be unified into the statement that the Picard group of the field is compact. This also relates to Dirichlet’s class number formula - the volume of the Picard group is hR, where h is the class number and R is the regulator.
The Picard group is a generalization of the ideal class group.
First, use characters to sort the primes into -functions.
Relate this to Cebotarev, Artin stuff.
Hard part is to show is non-zero for non-trivial characters.
Relates class numbers to zeta values.
Prototypical example is Riemann zeta function.
For class field theory, typically Dirichlet’s -functions are used These are used to prove Dirichlet’s theorem on primes in arithmetic progressions, and for his class number formula.
Modularity theorem tells us that the -function for an elliptic curves and the -function for the associated weight 2 Hecke-eigenform are identical.
Generalize to Weber -functions, where we use ray class groups.
Expected to satisfy functional equation/be identical for certain things - Langland’s program.
Expected to be zero-free in certain regions - Generalized Riemann hypothesis.
The asymptotic distribution of the prime numbers is .
The idea is that we count the primes using a special weighting, giving us better analytic properties. For example, Chebyshev’s function , where the Von-Mangoldt function is when is a prime power, and zero otherwise.
We relate this to the Riemann zeta function by looking at the logarithmic derivative
This can be transformed into the explicit equation , where .
The crux is to show that has no zeros on the line . This can be done using a trigonometric argument - launching point to pretentious theory. Once we have this, a careful accounting of possible types of zeros gives us the essence of the prime number theorem.
The remaining gap is solved using analysis, for example a Tauberian theorem.
If is an algebraic integer, then for some finitely generated additive subgroup , namely .
Conversely, let for some , where is a finitely generated subgroup . Let be a set of generators. Then , where is an matrix over . This means is an eigenvalue of , so , which is a monic polynomial with coefficients in . This is an algebraic integer.
Using the converse with the finitely generated subgroup , we get that the algebraic integers form a ring.
For a deg number field , is a degree -module.
Start w a -basis for . Clear denominators so that you have a -basis inside . Call the discriminant of this basis , and the basis itself .
We can write , with rational.
Consider the image of this under all the embeddings . If we solve for using Cramer’s rule, we get that , where , and is constructed like except we replace the th column by .
Thus, and are both algebraic integers, with . So , which implies is an integer. This shows that is contained in a free abelian group of rank .
Can be thought of as the square of the volume of the integral basis, int the lattice spaced generated from all the embeddings.
Is an invariant of a number field.
The ideal class group of a ring is finite.
The basic idea is to prove every ideal contains an element that is ‘small enough’ so something like where is independent of .
Here’s how we use this: take an ideal class , and consider an ideal . Use the above to find such an . So contains , therefore for some ideal .
Since , the ideal has ideal norm less than . Since is fixed, this bounds the possible divisors of , and so there are only a finite number of such ‘s. Since we can find such a for each ideal class, the ideal class group is finite!
Minkowski used a geometric argument to get a really nice . Every ideal class contains an ideal such that
To show this, we embed our ring of integers into as a lattice, using the different embeddings. The discriminant is the square of the volume of the fundamental parallelopiped.
An ideal will map to a sublattice, and the fundamental volume of this will be equal to times .
We can define a norm on this space that matches the standard norm. Now carefully construct a space where each element has norm has absolute value bounded by 1, which is compact, convex, and centrally symmetric, and has volume
Applying Minkowski’s convex body theorem to , which is scaled to have volume equal to (the volume of the fundamental parallelopiped for the lattice generated by ), we are guaranteed that there is a non-zero lattice point for inside .
This point gives us the that we need for the theorem.
Discriminant for prime case is .
Extension has degree over .
The Galois group is isomorphic to
Important special case for building up larger theorems in class field theory.
For a cyclotomic field , the Galois group over is cyclic of order . Let be the unique subfield of degree over , when .
splits into primes in , and is the order of mod . Since this is a cyclic group, the th powers form the unique subgroup of order .
Now observe that is the decomposition field for any prime over , since this is the only field over of degree .
Therefore, is equivalent to splitting completely in .
The inertial degree of over is defined as the degree of the extension over .
The decomposition group of over is The inertia group is , which is a normal subgroup of .
The quotient is a cyclic group of order - the inertial degree.
By Galois theory we can look at the fixed fields of and . If is a normal subgroup of , it turns out that has degree over , and splits into primes here. has degree over , and the primes of over have inertial degree here. Finally, has degree over , and the primes of over all have ramification index in .
This theorem gives a density for the prime ideals that split in a certain way (or alternatively, have an Artin symbol in a specific conjugacy class).
For an extension with Galois group , then the set of primes of , , and where , has Dirichlet density
The case when gives the density of the primes that split completely.
When is a cyclotomic extension of , this reduces to Dirichlet’s theorem on primes in arithmetic progressions.
The proof is important because it uses the technique of reducing to relatively cyclotomic extensions - used for the proof of Artin reciprocity.
For a ring with characteristic , there is an homomorphism called the Frobenius homomorphism taking to (and for the prime at infinity, it is complex conjugation). The basic question that reciprocity laws answer is ‘which ring homomorphism is it?’
For a prime field, every homomorphism is the identity - including the Frobeinus homomorphism. This is Fermat’s Little Theorem.
For a quadratic extension of a prime field with char , we could either get the identity or conjugation as the Frobenius homomorphism. We can look at this for all the primes (except for the ones dividing ) at once by studying .
In this case, Artin’s reciprocity law is that there exists a group homomorphism , which takes for any prime not dividing . This is equivalent to quadratic reciprocity.
More generally, suppose we have an abelian Galois extension , with the minimal polynomial of , and Galois group . Then there exists a group homomorphism which takes for primes not dividing .
This map is called the Artin map. It identifies primes with , which is the unique element of that extends the Frobenius map of . This also applies to the prime at infinity.
Even more generally, for , and a prime of over a prime of , then we define the Artin symbol to be the -power map modulo . This can be extended by multiplicativity to a homomorphism , which is the Artin map.
Another important fact is that the order of the Artin symbol is the inertial degree of over when is unramified. This implies that iff splits completely in . More generally, the Artin map tells you how factors in .
Let be an abelian extension, and let be a modulus divisible by all the primes of which ramify in (including infinite). Then
This gives a nice relation between generalized class groups and Galois groups, which can be used in particular for Hilbert class fields.
It’s more convenient to work with completions for all the primes at once, including the primes at infinity. Accounting for all the primes reveals a nice symmetry which can be used to state and prove the theorems of class field theory in a natural way.
We can define different absolute values on number fields, which must be multiplicative homomorphisms mapping 0 to 0, and satisfying a weakend form of the triangle inequality. We can arrange these into different equivalence classes based on the induced topology. These are called the palces. It turns out by Ostrowski’s theorem that each place corresponds to
For each place of a field we can construct the completion with respect to that place (analogously to the construction of from - which is a special case). We call this .
Let be the completion of at the place . We define as , where this is the restricted topological product. Also, , where are the elements with norm 1.
The major theorems of class field theory can be compactly stated as the following. There is an order reversing, bijective correspondence between the set of all finite abelian extensions , and the set of all open subgroups of which contain . Furthermore, .
Note that , the strict ray class group.
For another example, the functional equation for certain -functions can be elegantly proved using idelic theory, à la Tate’s thesis.
Generalization of ideal class group relative to a modulus .
Principal ideals relative to :
Ideal relative to : .
Ray class group is .
Strict if includes prime at infinity
Used to generalize Dirichlet’s Theorem on Primes in Arithmetic Progressions - Are there infinitely many prime ideals in each ray class? Weber -functions are defined for characters over Ray class groups.
The maximal unramified abelian extension of a number field (unramified at the infinite places also). It always exists.
A particularly important property is that its
Galois group is isomorphic to the ideal class group of (uniquely!). This gives it the ‘right’ to be called a class field, and allows us to use Galois theory and theorems from class field theory to prove results about the class group.
For example, we use this to prove Cebotarev’s density theorem.
For imaginary quadratic fields, the -invariant can be used to give you the Hilbert class field.
The inequalities tell us about the relation between and , where .
is an open subset of , so if is chosen so that , then
This generalizes the fact that for non-trivial characters which are trivial on .
If is a cyclic extension, and is divisible by a sufficiently high power of every ramified prime in , then .
Combining this with the Universal Norm Index inequality gives us equality.
Ideals in number fields have unique factorization into prime ideals. In algebraic number theory, the main focus is on how a prime ideal factors once it has been lifted up in a field extension.
In an extension of a number field, there are three different ways that primes in the base field factor.
They can split, ramify, or have inertia.
One way to see how a prime splits is to see how the minimal polynomial for the extension factors modulo that prime.
Say , and let be the minimal polynomial of over . Let be a prime not dividing , and let be a prime ideal of over . Then if factors mod as , we have that factors as . Furthermore, .
We can also use the Artin symbol to determine how an unramified prime splits. Consider an extension , along with a Galois extension containing , with Galois group . Call the Galois group of , . Fix a prime of lying over of , lying over a prime of . Then for each , consider the coset , and look at the orbits from right-action by the Artin symbol of over . The size of the orbit is the inertial degree of over , and thus the Artin symbol determines the splitting of in .
Take to be a normal extension.
Suppose that for all . Then by the C.R.T., there is a solution to the system of congruences and .
Let . Then , since .
On the other hand, , so . We can also write as the product of all , so this implies that , which is a contradiction.
The norm of an ideal of is . This is multiplicative, and the norm a the principal ideal is .
The norm of a prime above a prime is , where is the inertial degree.
Take a prime .
Now take ideal norms of everything. Since these are multiplicative, and since , the theorem follows.
When looking at an extension of number fields , a prime ramifies if for for some .
Let be a prime of lying over which ramifies. Then , with divisible by all the primes of lying over .
Take any integral basis. Replace one of the elements by . It is sufficient to show that divides the discriminant of this basis.
Assume the field is Galois (if not, use the Galois ext). For each automorphism, notice that is a prime lying over , so it contains . Thus, for all .
This means that contains the discriminant of this basis, and since the discriminant is an integer, it has to be in .