By Stein W.

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**Example text**

To employ our geometric intuition, as the Lenstras did on the cover of [LL93], it is helpful to view OK as a one-dimensional scheme X = Spec(OK ) = { all prime ideals of OK } over Y = Spec(Z) = {(0)} ∪ {pZ : p ∈ Z is prime }. There is a natural map π : X → Y that sends a prime ideal p ∈ X to p ∩ Z ∈ Y . For much more on this point of view, see [EH00, Ch. 2]. Ideals were originally introduced by Kummer because, as we proved last Tuesday, in rings of integers of number fields ideals factor uniquely as products of primes ideals, which is something that is not true for general algebraic integers.

Then I = p · p1 · · · pn , a contradiction. Thus every ideal can be written as a product of primes. Suppose p1 · · · pn = q1 · · · qm . If no qi is contained in p1 , then for each i there is an ai ∈ qi such that ai ∈ p1 . But the product of the ai is in the p1 · · · pn , which is a subset of p1 , which contradicts the fact that p1 is a prime ideal. Thus qi = p1 for some i. We can thus cancel qi and p1 from both sides of the equation. Repeating this argument finishes the proof of uniqueness. 10.

2 Q and Number Fields Magma has many commands for doing basic arithmetic with Q. 2. USING MAGMA 41 r4 + r1 > MinimalPolynomial(a+b); x^6 - 21*x^4 - 4*x^3 + 147*x^2 - 84*x - 339 > Trace(a+b); 0 > Norm(a+b); -339 There are few commands for general algebraic number fields, so usually we work in specific finitely generated subfields: > MinimalPolynomial(a+b); x^6 - 21*x^4 - 4*x^3 + 147*x^2 - 84*x - 339 > K := NumberField($1) ; // $1 = result of previous computation. > K; Number Field with defining polynomial x^6 - 21*x^4 - 4*x^3 + 147*x^2 - 84*x - 339 over the Rational Field We can also define relative extensions of number fields and pass to the corresponding absolute extension.