Download A Course In Algebraic Number Theory by Ash R.B. PDF

By Ash R.B.

It is a textual content for a uncomplicated direction in algebraic quantity thought.

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Example text

N−1 form an integral basis of B. Let p be a rational prime, and let f be the minimal polynomial of θ over Q. Reduce the coefficients of f modulo p to obtain f ∈ Z[X]. Suppose that the factorization of f into irreducible polynomials over Fp is given by f = he11 · · · herr . Let fi be any polynomial in Z[X] whose reduction mod p is hi . Then the ideal Pi = (p, fi (θ)) is prime, and the prime factorization of (p) in B is (p) = P1e1 · · · Prer . Proof. Adjoin a root θi of hi to produce the field Fp [θi ] ∼ = Fp [X]/hi (X).

N , with σr1 +j paired with its complex conjugate σr1 +r2 +j , j = 1, . . , r2 . Thus there are 2r2 complex embeddings, and r1 + 2r2 = n. The canonical embedding σ : L → Rr1 × Cr2 = Rn is the injective ring homomorhism given by σ(x) = (σ1 (x), . . , σr1 +r2 (x)). 2 Some Matrix Manipulations Let x1 , . . , xn ∈ L be linearly dependent over Z (hence the xi form a basis for L over Q). Let C be the matrix whose k th column (k = 1, . . , n) is σ1 (xk ), . . , σr1 (xk ), Re σr1 +1 (xk ), Im σr1 +1 (xk ), .

Consequently, g [B/P B : A/P ] = i=1 ei fi , as claimed. To prove the second equality, we first assume that B is a free A-module of rank n. 8), this covers the case where A is a PID, in particular, when L is a number field. If x1 , . . , xn is a basis for B over A, we can reduce mod P B to produce a basis for B/P B n over A/P , and the result follows. Explicitly, suppose i=1 (ai +P )(xi +P B) = 0 in B/P B. n Then i=1 ai xi belongs to P B, hence can be written as j bj yj with bj ∈ B, yj ∈ P . Since bj = k cjk xk with cjk ∈ A, we have ak = j cjk yj ∈ P for all k.

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