Proposition 3

Tag 06LF, permanent. Cite it as [Bourbaki, Tag 06LF].

Algebra, chapter V, § 8. Norms and traces

— Let E be an extension of finite degree n of K and x an element of E, of degree d over K. Write $f(X) = X^d + \sum_{i=1}^d a_i X^{d-i}$ for the minimal polynomial of X over K. Then we have

\text{Tr}_{E/K}(x) = - \frac{n}{d} a_1 \\ \text{N}_{E/K}(x) = (((-1)^d a_d)^{n/d} = (-1)^n a_d^{n/d} .

Prop. 3 follows directly from the Cor. of Prop. 2 and the lemma:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Lemma 3, § 8

Cites

Cor. of Prop. 2