# Integral domain.

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

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 $b=0)$.
• An element of $A$ is $\alpha + x^2. f(x)$ when $f\in k[x]$ – Ali NoumSali Traore Aug 6 '18 at 20:06
• Yes and such an element is a polynomial in $k[x]$. – paf Aug 6 '18 at 20:07
• Isn't $(x^2)$ a prime ideal of $A$? Then the quotient $A/(x^2)$ is...? – misogrumpy Aug 6 '18 at 20:36