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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.

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    $\begingroup$ Is this not what it means that $E$ is a finitely generated $k$-algebra? $\endgroup$
    – Jyrki Lahtonen
    Jul 8, 2015 at 20:39
  • $\begingroup$ What is $F$ in this problem? But yes, that opening line merely repeats that $E$ is a finitely-generated $k$ algebra. $\endgroup$
    – hardmath
    Jul 8, 2015 at 20:43

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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 $k$.

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