Library / Algebra / Chapter III / § 9
Corollary
Tag 0662, permanent. Cite it as [Bourbaki, Tag 0662].
Algebra, chapter III, § 9. Norms and traces
Let A be a commutative K-algebra admitting a finite basis over K and B an A-algebra admitting a finite basis over A. Then B admits a finite basis over K, and, for all b ∈ B b \in B b ∈ B
Tr B / K ( b ) = Tr A / K ( Tr B / A ( b ) ) , N B / K ( b ) = N A / K ( N B / A ( b ) ) P C B / K ( b ; X ) = N A [ X ] / K [ X ] ( P C B / A ( b ; X ) ) .
\begin{align*}
\operatorname{Tr}_{B/K}(b) &= \operatorname{Tr}_{A/K}(\operatorname{Tr}_{B/A}(b)), \quad N_{B/K}(b) = N_{A/K}(N_{B/A}(b)) \\
P_{C_{B/K}}(b; X) &= N_{A[X]/K[X]}(P_{C_{B/A}}(b; X)).
\end{align*}
Tr B / K ( b ) P C B / K ( b ; X ) = Tr A / K ( Tr B / A ( b )) , N B / K ( b ) = N A / K ( N B / A ( b )) = N A [ X ] / K [ X ] ( P C B / A ( b ; X )) . (26) This follows immediately from Proposition 6, setting
V = B V = B V = B and
u ( x ) = b x u(x) = bx u ( x ) = b x .
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Cites
Transcribed from the printed edition, 1998, Springer. Not translated.