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Let $L/K$ be a finite extension of number fields. $\frak P$ denotes a principal prime ideal of $\mathcal O_L$. Then I want to show that $N_{L/K}(\mathfrak P)$ is principal. The norm is defined as $$ N_{L/K}(\mathfrak{P})= \mathfrak{p}^{f (\mathfrak P|\frak p)}, $$ where $\mathfrak p =\mathfrak P \cap \mathcal O_K$.

I know that the contraction of a principal ideal may not be principal, so I can't work it out by showing $\mathfrak p$ is principal.

In fact, the conclusion is used in Introduction to Cyclotomic Fields by Washington on page 7 at the last paragraph. See also this link. The original exercise assigned by my professor is finding the unique prime ideal $\mathcal P$ of $L:=\mathbb Q(\zeta_{23})$ above $\mathfrak p= (2,(1+\sqrt{-23})/2)$ of $K=\mathbb Q(\sqrt{-23})\subset \mathbb Q(\zeta_{23})$.

I am a beginner of ANT so I am sorry for asking the possibly trivial question. Any help would be appreciated.

zc l
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  • Borevich/Shafarevich's Number Theory develops the norm mapping on ideals with the requirement that it behave as you want on principal ideals: ${\rm N}{L/K}(\alpha\mathcal O_L) = {\rm N}{L/K}(\alpha)\mathcal O_K$. See Theorem 4 on p. 196 and the following few pages, esp. equations (5.11) and Theorem 5 on p. 200. – KCd Nov 16 '22 at 02:36
  • @KCd Thanks for the reference! – zc l Nov 16 '22 at 06:56

1 Answers1

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Yes. It is generated by the norm of a generator. (If L/K is a field extension, the norm of $\alpha \in L^\times$ is the determinant of the multiplication-by-$\alpha$ map (which is a $K$-linear endomorphism of $L$ viewed as a $K$-vector space). When $L/K$ is Galois, this is just the product of the Galois conjugates.)

hunter
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  • Thank you for your answer. I think you mean that the norm of an ideal can be defined as the ideal generated by the norms of all its elements. If so, then my question is clear. I find the conclusion in Algebraic Number Fields by Janusz. Thank you very much again. – zc l Nov 12 '22 at 16:17