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Let $A=k[x_1,x_2]/(x_2^2-x_1^3+x_1)$. As an example of Noether normalization, determine elements $y_1,\ldots,y_m\in A$, algebraically independent over $k$, such that $A$ is a finite $k[y_1,\ldots,y_m]$-algebra.

This is a problem in the Klaus Hulek's Elementary Algebraic Geometry. I think the book's proof of Noether normalization is not actually constructive...

Could anyone show me how to determine the $y_1,\ldots,y_m$ ?

user26857
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Jacob Ralph
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  • Well, it doesn't say "application", it says "example. Have you tried some things? Have you looked at simpler examples? Here's the question that should always be asked first: What is the simplest finitely generated $k$-algebra for which you cannot calculate a noether normalization? – Andrew Dudzik Oct 19 '14 at 05:03
  • Oh, and one more question, perhaps an important one: How large do you think $m$ is, in this example? – Andrew Dudzik Oct 19 '14 at 05:05
  • I didn't encounter any example of this kind before...so I have got no clue to how I may find such y1~ym. Certainly, m must be lower than 3. Could you give any hint? – Jacob Ralph Oct 19 '14 at 07:40
  • If you know this basic theorem relating the dimension to transcendence degree, you can determine $m$ immediately, assuming that you know the Krull dimension of $A$. – Andrew Dudzik Oct 19 '14 at 07:42
  • Honestly, almost anything you try will lead to progress here. Like I said if the problem is too hard, try to do it for something with one variable. – Andrew Dudzik Oct 19 '14 at 07:43
  • The proof of Noether normalization which can be found in almost every book on commutative algebra is constructive. It can be used to find specific elements as required. – Martin Brandenburg Oct 19 '14 at 07:55

1 Answers1

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The Krull dimension of $A$ is $1$, so almost every choice of $y\in A-k$ is good.

user26857
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