Subring of polynomials Let $k$ be a field and $A=k[X^3,X^5] \subseteq k[X]$.
Prove that:
a. $A$ is a Noetherian domain.
b. $A$ is not integrally closed.
c. $dim(A)=?$ (the Krull dimension).
I suppose that the first follows from $A$ being a subring of $k[X]$, but I don't know about the rest.
Thank you in advance.

a) Not every subring of a noetherian ring is noetherian (there are plenty of counterexamples), so this doesn't work here. Instead, use Hilbert's Basis Theorem.
b) The element $X^2 = \frac{X^5}{X^3}$ is in $\mathrm{Quot}(A)$. Try to show that it is integral over $A$, but not in $A$.
c) The dimension is the transcendence degree of $\mathrm{Quot}(A)$ over $k$. But this field is easy to compute.
