- Is $\mathbb{Z}[\sqrt{17}]$ a Noetherian domain?
- Is $\mathbb{C}[x^2, x^3]$ a Dedekind domain?
Can't seem to make any headway with these. Any help appreciated
Can't seem to make any headway with these. Any help appreciated
$\mathbb{Z}[\sqrt{17}]$ is isomorphic to the quotient of $\mathbb{Z}[x]$ modulo the ideal $(x^2-17)$. Is $\mathbb{Z}[x]$ Noetherian? Is a quotient of a Noetherian ring a Noetherian ring?
Dedekind domains are integrally closed; can you find a monic polynomial with coefficients in $\mathbb{C}[x^2,x^3]$ that has solutions in $\mathbb{C}(x^2,x^3)$ but not in $\mathbb{C}[x^2,x^3]$?
$\mathbb {Z}[\sqrt{17}]$ is a Noetherian ring, as $\mathbb Z$ is clearly Noetherian, so by Hilbert's basis theorem $\mathbb Z[x]$ is, and since the homomorphic image of a Noetherian ring is Noetherian we have that $\mathbb{Z}[x]/(x^2-17)\cong \mathbb Z[\sqrt{17}]$ is Noetherian. Additionally, $(x^2-17)$ is a prime ideal (why?), so $\mathbb {Z}[\sqrt{17}]$ is a domain.
$\mathbb C[x^2,x^3]$ is not integrally closed, so not a Dedekind domain. I'll let you find an example of an element integral over it but not in it.