By Ash R.B.

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

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A Course In Algebraic Number Theory

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1), represent elements of B by matrices with respect to an integral basis ω1 , . . , ωn of B. Reduction of the entries mod p gives matrices representing elements of B/(p). 2. Show that a nilpotent element (or matrix) has zero trace. 3. A PRACTICAL FACTORIZATION THEOREM 7 Suppose that A(β), the matrix representing the element β, is nilpotent mod p. Then A(βωi ) will be nilpotent mod p for all i, because βωi is nilpotent mod p. 3. By expressing β in terms of the ωi and computing the trace of A(βωj ), show that if β is nilpotent mod p and β ∈ / (p), then d ≡ 0 mod p, hence p divides d.

To prove this, assume g(θ) ∈ ker λi . With a 8 CHAPTER 4. FACTORING OF PRIME IDEALS IN EXTENSIONS subscript 0 indicating reduction of coefficients mod p, we have g0 (θi ) = 0, hence hi , the minimal polynomial of θi , divides g0 . If g0 = q0 hi , then g − qfi ≡ 0 mod p. Therefore g(θ) = [g(θ) − q(θ)fi (θ)] + q(θ)fi (θ) so g(θ) is the sum of an element of (p) and an element of (fi (θ)). Thus ker λi ⊆ Pi , so Pi = ker λi , a prime ideal. We now show that (p) divides P1e1 · · · Prer . We use the identity (I+I1 )(I+I2 ) ⊆ I+I1 I2 , where I, I1 and I2 are ideals.

Let S be the set of all a1 e1 + a2 e2 with −1 < ai < 1, i = 1, 2. Then µ(S) = 4v(H), but S contains no nonzero lattice points. Thus compactness is a necessary hypothesis in part (b). 2. 2 3 A Volume Calculation We will use n-dimensional integration technique to derive a result that will be needed in the proof that the ideal class group is finite. We will work in Rn , realized as the product of r1 copies of R and r2 copies of C, where r1 + 2r2 = n. Our interest is in the set r1 Bt = {(y1 , . . , yr1 , z1 , .

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


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