Proposition 3

Tag 00KL, permanent. Cite it as [Bourbaki, Tag 00KL].

Algebra, VIII, p. 340, § 17. Reduced Norms and Traces

An element $a$ of A is invertible if and only if its reduced norm is nonzero. In particular, A is a field if and only if we have Nrd $_{A/K}(a)\not= 0$ for every nonzero element $a$ of A.

An element $a$ of A is invertible if and only if its norm is nonzero (III, §9, No. 4, p. 545, Proposition 3). Proposition 3 therefore follows from formula (24).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Remark, § 17
Theorem 2 (Wedderburn), § 18

Cites

III, §9, No. 4, p. 545, Proposition 3 not in this corpus