1
$\begingroup$

Let k a field.

Let $ A = k+ x^2 k[x] $,

Show that A is integral domain and finite type.

An element of $A$ is $\alpha + x^2. f(x) $ where$ f\in k[x] $

I know that A is a subalgebra of k[x], Any hint for integral domain

$\endgroup$
2
$\begingroup$

Since $A\subset k[x]$ and since $k[x]$ is an integral domain, we can see that for all $a,b\in A$, $(ab = 0$ in $A) \Rightarrow (ab = 0$ in $k[x]) \Rightarrow (a=0$ or $b=0)$.

$\endgroup$
  • $\begingroup$ An element of $A$ is $\alpha + x^2. f(x) $ when $ f\in k[x] $ $\endgroup$ – Ali NoumSali Traore Aug 6 '18 at 20:06
  • $\begingroup$ Yes and such an element is a polynomial in $k[x]$. $\endgroup$ – paf Aug 6 '18 at 20:07
  • $\begingroup$ Isn't $(x^2)$ a prime ideal of $A$? Then the quotient $A/(x^2)$ is...? $\endgroup$ – misogrumpy Aug 6 '18 at 20:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.