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Math 129 - Number Fields Math 129 - Number Fields Taught by Barry Mazur Notes by Dongryul Kim Spring 2016 This course was taught by Barry Mazur. We met twice a week on Tues- days and Thursdays from 10:00 to 11:30 in Science Center 507. We used the textbook Number fields by Daniel M. Marcus. There were 11 students taking the course. There was an in-class final exam, and the course also required one thirty-minutes-long presentation. The course assistant was Kevin Yang. Contents 1 January 26, 2016 4 1.1 Algebraic numbers and integers . .4 1.2 Quadratic fields . .6 2 January 28, 2016 7 2.1 Gauss's lemma . .7 2.2 Primitive element theorem . .8 2.3 Roots of unity . .8 3 February 2, 2016 10 3.1 The fundamental embedding . 10 3.2 Trace and norm . 11 3.3 Galois size and S-numbers . 12 4 February 4, 2016 13 4.1 Integral closure . 13 4.2 Fundamental embedding to K ⊗ R ................. 13 4.3 Weil numbers . 14 4.4 Fermat's last theroem . 15 5 February 9, 2016 16 5.1 Discriminant . 16 6 February 11, 2016 19 6.1 Reasons for loving the discriminant . 19 6.2 S-numbers . 20 1 Last Update: August 27, 2018 7 February 16, 2016 21 7.1 Dedekind domain . 21 7.2 Factorization of ideals . 22 8 February 18, 2016 25 8.1 Order of an ideal . 25 8.2 Finite approximation . 26 8.3 Residue fields . 26 8.4 Unique factorization implies principal ideal domain . 27 9 February 23, 2016 28 9.1 Quotient of ideals . 28 9.2 Ramification indices and residue field degrees . 29 9.3 Spectrum of a ring . 30 10 February 25, 2016 32 10.1 Decomposition equation . 32 10.2 Ramification and the discriminant . 33 10.3 The Kronecker-Weber theorem . 34 11 March 1, 2016 35 11.1 The decomposition group . 35 11.2 The inertia group . 36 12 March 3, 2016 38 12.1 Ramification in the tower . 38 12.2 Ideal class group of Q[ζ23]...................... 39 13 March 8, 2016 42 13.1 Frobenius structure of a Galois group . 42 14 March 10, 2016 44 14.1 Finiteness of the ideal class group . 44 15 March 22, 2016 46 15.1 The Minkowski bound . 46 16 March 24, 2016 48 16.1 Computation of ideal class group . 48 16.2 Higher ramification groups . 49 17 March 29, 2016 51 17.1 Logarithm of the fundamental embedding . 51 18 March 31, 2016 53 18.1 Dirichlet unit theorem . 53 18.2 The different ideal . 54 2 19 April 5, 2016 56 19.1 Counting ideals in ideal classes . 57 20 April 7, 2016 59 20.1 Equidistribution of ideals in ideal classes . 59 20.2 The Chebotarev density theorem . 60 21 April 12, 2016 61 21.1 Distribution of ideals in the class group . 61 21.2 The regulator . 62 22 April 14, 2016 64 22.1 Dirichlet Series . 64 22.2 Minkowski's thereom . 65 23 April 19, 2016 66 23.1 The L-function . 66 23.2 Extending the zeta function . 67 24 April 21, 2016 69 24.1 Infinite products . 69 24.2 Density of primes . 70 24.3 Odlyzko's bound . 70 25 April 26, 2016 72 25.1 Polar density . 72 25.2 A density theorem . 73 25.3 Cyclotomic units . 73 A The Chebotarev density theorem 75 A.1 The Frobenius element . 75 A.2 The Chebotarev density theorem . 75 A.3 Consequences . 76 3 Math 129 Notes 4 1 January 26, 2016 There will be half-hour presentations. I have experimented this in Math 124, and it worked well. 1.1 Algebraic numbers and integers Definition 1.1. An algebraic number is a root of a (monic) polynomial d d−1 f(X) = X + ad−1X + ··· + a0 for ai 2 Q. An algebraic integer is a root of a monic polynomial with integer coefficients. These have connection with many branches of mathematics. We can look at the ring A of algebraic integers and talk about Spec A, which we will discuss. This A can be viewed as a lattice in the Euclidean space thus a geometric structure. For instance, the ring p p Z[ −D] = fa + b −D : a; b 2 Zg on the complex plane looks like a rectangular lattice. There are also connections with analysis; the zeta function contains information about the nature of prime ideals in A. Proposition 1.2. An algebraic integers that is a rational number is an integer. m Proof. Let n be an algebraic integer for (m; n) = 1. Then there will be integers ad−1; ad−2; : : : ; a0 such that md md−1 + a + ··· + a = 0: n d−1 n 0 Then multiplying nd to both sides, we get d d−1 d m + ad−1nm + ··· + a0n = 0 and thus n divides m. This means that n = ±1. Proposition 1.3. If α is an algebraic number then there is an integer N ≥ 1 such that Nα is an algebraic number. Proof. There is are integers ai and N 6= 0 that makes α a root of the polynomial a a a f(X) = Xd + d−1 Xd−1 + d−2 Xd−2 + ··· + 0 = 0: N N N Then d d d−1 d−1 d−2 d−2 d N X + ad−1N X + ad−2N · N X + ··· + N a0 = 0 and thus Y = Nα is the root of d d−1 d−2 d F (Y ) = Y + ad−1Y + ad−2NY + ··· + N a0 = 0: Math 129 Notes 5 Proposition 1.4. Let z1; z2 be two complex numbers that are linearly indepen- dent over Q. Suppose that ( αz1 = a11z1 + a12z2 αz2 = a21z1 + a22z2 for some rational numbers a11; a12; a21; a22. Then α is an algebraic number. Proof. Let a11 a12 A = 2 Mat2(Q): a21 a22 Then we see that det(α · I2 − A) = 0 and hence α is a root of the characteristic polynomial X2 − tr(A)X + det(A): This completes the proof. Note that if the aijs were integers, then α should have been an algebraic integer. We can generalize it to more variables. In fact, we have the following claim. Proposition 1.5. Let V ⊂ C be a finite-dimensional Q-vector space, and let M ⊂ C be a finitely generated abelian group. Consider an α 2 C. If α · V ⊆ V , then α is an algebraic number. If α · M ⊆ M, then α is an algebraic integer. Actually we need to use the fundamental theorem of finitely generated abelian groups. Since M is finitely generated and torsion-free (M is a sub- set of C), it has to be isomorphic to some Zr. Then we can run the exactly same argument for the previous proposition. Also note that the converse is also true. Given an algebraic number/integer α 2 C, we can set V /M to be the vector space/abelian group generated by the powers of α. Corollary 1.6. Let K=Q be a field extension of finite degree. Then every ele- ment of K is an algebraic number. Proof. Just set V = K. Theorem 1.7. The sum and product of two algebraic numbers are again alge- braic numbers. The sum and product of two algebraic integers are agin algebraic integers. Proof. This is because Q[α; β] is a finitely generated vector space over Q, and Z[α; β] is a finitely generated abelian group. Math 129 Notes 6 1.2 Quadratic fields For a extension field K over Q, we denote by AK the right of algebraic integers in K. Let us try to describe the structure of AK for a extension field K over Q of degreep 2. Consider any α 2 K. By the quadratic formula,p we know that α = r + s · D for some square-free D 2 Z. Letα ¯ = r − s · D be the Galois conjugate. Then α is the root of the quadratic polynomial X2 − 2rX + (r2 − Ds2) = 0: It follows from this fact that α is an algebraic if and only if both 2r and r2 −Ds2 are integers. In fact, we can more explicitly describe the additive group of algebraic integers. Exercise 1.8. Assume that K is a extension field over Q of degree 2. Then the pring of algebraic integers AK is either (i) the abelian groupp generated by 1 and D or (ii) the abelian group generated by 1 and (1 + D)=2. Math 129 Notes 7 2 January 28, 2016 There is a lemma I would like to begin with: Lemma 2.1. Let K be a field and let f(X) 2 K[X] be a polynomial. If f(X) and f 0(X) are relatively prime, then f(X) has no multiple roots. Let θ 2 C be an algebraic number. Then there is a unique irreducible monic polynomial fθ(X) 2 Q[X] that has θ as a root with smallest degree d. We can write fθ(X) = (X − θ1) ··· (X − θd): By the above lemma and the fact that f and f 0 has to be relatively prime, we see that θ1; θ2; : : : ; θd are all distinct. We call the set fθ1; : : : ; θdg the full set of Galois conjugates. If θ and θ0 are Galois conjugates, or in other words, roots of the same irre- ducible polynomial over Q, then we have a natural isomorphism Q[θ] =∼ Q[θ0] Q 2.1 Gauss's lemma Factorization in Z[X] and Q[X] might be slightly different.
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