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Let $g(x)=(x-a)(x-b)(x-c)$ with $a\neq b\neq c$. Then $\mathbb{C}[x,y]/\langle y^2-g(x) \rangle$ is integrally closed in its fraction field.

I manage to do with the following theorem:

A smooth function $f$ is non-singular if and only if $\mathbb{C}[x,y]/\langle f(x,y) \rangle$ is integrally closed in its fraction field.

But how will I do without using this? Like an example has done here.

XYZABC
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  • What is the minimal polynomial of $\frac{u_1(x)}{v_1(x)}+g(x)^{1/2}\frac{u_2(x)}{v_2(x)}$ over $\mathbb{C}(x)$ ? – reuns Nov 25 '17 at 04:27
  • If I can show it is UFD, then it is solved. But why it is UFD? – XYZABC Nov 25 '17 at 04:27
  • @reuns why we want to find minimal polynomial of $\frac{u_1(x)}{v_1(x)}+g(x)^{1/2}\frac{u_2(x)}{v_2(x)}$ – XYZABC Nov 25 '17 at 04:29
  • To find when it is integral over $\mathbb{C}[x]$, of course. And what does "monic polynomial" mean here ? – reuns Nov 25 '17 at 04:29

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