By Ash R.B.

This can be a textual content for a easy path in algebraic quantity idea.

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Completing the induction. Note that n + 2 (rather than n + 1) is correct, because r1 + 2(r2 + 1) = r1 + 2r2 + 2 = n + 2. 1 The Canonical Embedding Deﬁnitions and Comments Let L be a number ﬁeld of degree n over Q, and let σ1 , . . , σn be the Q-monomorphisms of L into C. If σi maps entirely into R, we say that σi is a real embedding; otherwise it is a complex embedding. Since the complex conjugate of a Q-monomorphism is also a Qmonomorphism, we can renumber the σi so that the real embeddings are σ1 , .

Let P1 , . . 1). Then a ∈ I, so (a) ⊆ I. 5)], I divides (a), so (a) = II for some nonzero ideal I . If I is divisible by Pi , then I = Pi I0 for some nonzero ideal I0 , and (a) = IPi I0 . 1). 3 Corollary A Dedekind domain with only ﬁnitely many prime ideals is a PID. Proof. Let J be the product of all the nonzero prime ideals. 2) there is a nonzero ideal I such that II is a principal ideal (a), with I relatively prime to J. But then the set of prime factors of I is empty, so I = R. Thus (a) = II = IR = I.

Diﬀerentiate the equation (X p r−1 r − 1)Φpr (X) = X p − 1 to get r−1 (X p r−1 − 1)Φpr (X) + pr−1 X p −1 Φpr (X) = pr X p r −1 . Setting X = ζ and noting that ζ is a root of Φpr , we have (ζ p r −1 − 1)Φpr (ζ) + 0 = pr ζ p r −1 . Thus pr ζ p −1 . 6). The norm of ζ is ±1, as r r−1 ζ is a root of unity. The norm of pr is prϕ(p ) = prp (p−1) . 3), the norm is multiplicative, so the norm of Φpr (ζ) is ±pc , where c = r(p − 1)pr−1 − pr−1 = pr−1 (pr − r − 1). 5), we related the norm of an ideal I to the ﬁeld discriminant d and the discriminant D(z) of a basis z for I.