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Algebraic Number Theory TIFR VSRP Programme Project Report Algebraic Number Theory Milind Hegde Under the guidance of Prof. Sandeep Varma July 4, 2015 ACKNOWLEDGMENTS I would like to express my thanks to TIFR for selecting me to participate in the Vis- iting Students’ Research Programme 2015, which has been a wonderful experience. I would also like to thank Prof. Sandeep Varma for being my guide during this project and teaching me many aspects of algebraic number theory, without which I would not have been able to create this report. 2 C O N T E N T S 1 algebraic number theory 4 1.1 Tensor Products . 4 1.2 Norm, Trace, and Discriminant . 5 1.3 The Ring of Integers . 6 1.4 Dedekind Domains . 8 1.4.1 Unique Factorization of Ideals . 8 1.4.2 Fractional Ideals and the Ideal Class Group . 10 1.4.3 Factorization of Primes in Extensions . 11 1.5 Finiteness of Class Number . 11 3 A L G E B R A I C N U M B E R T H E O RY 1.1 tensor products Definition. (Tensor Product). Given two modules M and N o v e r a ring R, w e can define a tensor product of the modules, denoted M ⊗ N, which comes equipped with a bilinear map ⊗ : M × N ! M ⊗ N and satisfies the following universal property: Given a bilinear map f : M × N ! M0, there exists a unique linear map fe: M ⊗ N ! M0 such that f(m, n) = f˜(m ⊗ n) for all (m, n) 2 M × N, which is to say that the following diagram commutes: ⊗ M × N M ⊗ N f fe M0 The tensor product of any two R-modules exist, and it can be constructed as a quotient of the free module with basis of all formal symbols m ⊗ n, (m, n) 2 M × N b y appropriate relations. The tensor product respects direct sums, i.e. M0 ⊗ (M ⊕ N) = (M0 ⊗ M) ⊕ (M0 ⊗ N). W e must also define the tensor product of maps. Given maps f : M ! M0 and g : N ! N0, w e have a map f ⊗ g : M ⊗ N ! M0 ⊗ N0, which w e obtain as the unique map satisfying the universal property of M ⊗ N, when applied to the bilinear map (m, n) 7! f(m) ⊗ g(n) from M × N to M0 ⊗ N0. The tensor product is right exact, which means given a short exact sequence 0 ! N0 ! N ! N00 ! 0, w e can get a new exact sequence b y tensoring with a R-module M, namely M ⊗ N0 ! M ⊗ N ! M ⊗ N00 ! 0. The maps in the new exact sequence are the corresponding maps in the original, tensored with the identity map on M. (Saying the tensor product is right exact is the same as saying that the tensoring of R-linear maps preserves surjectivity.) W e will use the following proposition several times: Proposition. Let k be a field, Ω be an extension of k, and f 2 k[X] a monic irreducible polynomial. Then, ∼ k[X]=(f(X)) ⊗k Ω = Ω[X]=(f(X)) 4 ∼ Proof. First we will prove that k[X] ⊗k Ω = Ω[X]. Consider the following ring homomorphisms g : k[X] ⊗k Ω ! Ω[X], p(X) ⊗ ! 7! !p(X) n n h : Ω[X] ! k[X] ⊗k Ω, anX + ::: + a0 7! X ⊗ an + : : : 1 ⊗ a0 It can be easily checked that g ◦ h and h ◦ g are identity maps on the respective rings, and so the two rings are isomorphic. Now consider the short exact sequence below: 0 ! (f(X)) ! k[X] ! k[X]=(f(X)) ! 0. Tensoring this with Ω, we get (using the above isomorphism) (f(X)) ⊗k Ω ! Ω[X] ! k[X]=(f(X)) ⊗k Ω ! 0. Hence k[X]=(f(X)) ⊗k Ω is isomorphic to Ω[X] quotiented by the image of (f(X)) ⊗k Ω in Ω[X]. But this can easily be seen to be (f(X)): any element of (f(X)) in Ω[X] is of n the form g(X)f(X) for some g 2 Ω[X]. Writing g(X) = gnX + ::: + g0, we see that the n image of X f(X) ⊗ gn + ::: + f(X) ⊗ g0 2 (f(X)) ⊗k Ω is g(X)f(X). 1.2 norm, trace, and discriminant Definition. (Norm and Trace). Suppose L=K is a finite field extension and that α 2 L. Then multiplication by α is a linear map from L to L. The determinant and trace of the map Tα : L ! L, x 7! αx are well-defined, and are respectively called the norm and trace of α (and denoted NmL=K(α) and TrL=K(α)). From the definitions it clearly follows that the norm and trace are respectively multiplicative and additive, i.e. Nm(αβ) = Nm(α) Nm(β) and Tr(α + β) = Tr(α) + Tr(β). Next we begin defining the important quantity called the discriminant, which has many formulations, depending on the context. Definition. (Discriminant). • If V is a finite dimensional vector space over a field F and : V × V ! F is a bilinear form, then the discriminant of , with respect to a basis feig of V, is defined as det( (ei, ej)). If we find the discriminant with respect to a different basis, say ffig, then we have that 2 det( (fi, fj)) = det(aij) det( (ei, ej)),(1) where fj = ajiei. A bilinear formP for which the discriminant is non-zero with respect to some basis (and hence any basis, by (1)) is said to be non-degenerate. 5 • For a finite field extension L=K, the discriminant of the extension is defined as the discriminant of the bilinear form (α, β) 7! TrL=K(αβ) (regarding L as a vector space over K). From (1), we see that this quantity is well-defined upto 2 multiplication by the square of an element of K, i.e. it is an element of K×=K× . 1.3 the ring of integers For an extension K of Q, we want to find a subring of K which will play the same role as what Z plays with Q. To this end, we define when an element is said to be integral. Definition. (Integral Elements). Let A be a subring of a field K. Then an element α 2 K is said to be integral over A if it satisfies a monic polynomial with coefficients in A. First we verify that the only elements of Q integral over Z lie in Z, to ensure that p the definition is really what we want. So suppose q 2 Q with (p, q) = 1 satisfies n n-1 a monic polynomial with integral coefficients, say X + an-1X + ::: + a1X + a0. Substituting and clearing the denominators, we get n n-1 n-1 n p + an-1p q + ::: + a1pq + a0q = 0 Now if q 6= 1, let π be a prime dividing q and not p. The above equation implies π j pn =) π j p, a contradiction. So q = 1, as required. The observation that the set of elements of Q integral over Z equals Z leads us to the following definition. Definition. (Integrally Closed). An integral domain A is said to be integrally closed if the set of elements of its fraction field K that are integral over A equals A itself. The above argument directly extends to any UFD, and so we get the following proposition: Proposition. Any UFD is integrally closed. These observations lead us to suspect that the set of integral elements over an integral domain will form a ring (which agrees with our prototypical example of Z), which we prove next. Theorem 1. If L is a field and A ⊂ L is a subring (and hence an integral domain), then the set of integral elements of L over A forms a ring. This ring is denoted OL and is called the ring of integers. The majority of the proof lies in an important lemma which will be useful later as well. Lemma. An element α 2 L is integral over A if and only if there exists a finitely generated A-submodule M of L such that αM ⊆ M. 6 Proof. If α is integral, there exists a monic polynomial with coefficients in A that it satisfies, i.e. for some ai 2 A, n n-1 α + an-1α + ::: + a1α + a0 = 0 This implies that M = A[α] is finitely generated, and clearly αM ⊆ M. Now suppose we have a finitely generated A-submodule M ⊆ L with αM ⊆ M. Then consider the linear map Tα : M ! M, m 7! αm, which is well-defined since αM ⊆ M. Let e1, ::: , en be a generating set of M. Then there exist aij 2 A (not necessarily unique) such that αei = aijej, j = 1, ::: , n j X Bringing the right side to the left, we get (α - a11)e1 - a12e2 - ::: - a1nen = 0 -a21e1 + (α - a22)e2 - ::: - a2nen = 0 . -an1e1 - an2e2 - ::: + (α - ann)en = 0 Regarding this as a system of equations with coefficients in L, we see that it has a non-trivial solution, since all the ei 2 L cannot be zero. Hence if the matrix of coefficients is A, we have that det A = 0, which on expanding gives a monic polynomial in α with coefficients in A. Proof of Theorem 1. We are given α, β integral over A. By the lemma we have finitely generated A-submodules of L, M and N, such that αM ⊆ M and βN ⊆ N. The submodule k MN = mini j mi 2 M, ni 2 N, k 2 N i=1 X is also finitely generated and it is clear that (αβ)MN ⊆ MN and (α + β)MN ⊆ MN, and so by the lemma we have that α + β and αβ are integral over A, as required.
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