Atiyah-MacDonald: proof of Proposition 7.9, weak Nullstellensatz.

Question

Proposition 7.9 in Atiyah & MacDonald's Introduction to Commutative Algebra states:

Let $k$ be a field and $E$ a finitely-generated $k$ algebra. If $E$ is a field, then it is finite algebraic extension of $k$.

The proof begins with the line: ''Let $E =k[x_1, \dots, x_n]$". Would someone be able to explain why this is obviously always possible? It seems that we are assuming something we are trying to prove.

Is this not what it means that $E$ is a finitely generated $k$-algebra? — Jyrki Lahtonen, Jul 08 '15 at 20:39
What is $F$ in this problem? But yes, that opening line merely repeats that $E$ is a finitely-generated $k$ algebra. — hardmath, Jul 08 '15 at 20:43

score 2 · Answer 1 · answered Jul 08 '15 at 20:41

2

There are a finite number of generators for $E$ as a $k$ algebra - which is what the statement expresses.

The substantial content of the proposition is that if $E$ is a field then the $x_i$ (and hence all elements of $E$) are algebraic over $k$.

answered Jul 08 '15 at 20:41

Mark Bennet

100,194

Atiyah-MacDonald: proof of Proposition 7.9, weak Nullstellensatz.

1 Answers1