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.