Name a field ($\ne \Bbb{R}$ or $\Bbb{C}$) which contains a root of $x^5+2x^3+4x^2+6$.

Question

This question originates from Pinter's Abstract Algebra, Chapter 27, D6.

Name a field ($\ne \Bbb{R}$ or $\Bbb{C}$) which contains a root of $x^5+2x^3+4x^2+6$.

Suppose $p(x)=x^5+2x^3+4x^2+6$ is a polynomial over $\Bbb{Q}$. Let $c$ be a root of $p(x)$. Note $p(x)$ is irreducible by Eisenstein's Criteria. The smallest field containing $\Bbb{Q}$ and $c$ is $\Bbb{Q}(c)$ which is isomorphic to the quotient field $\Bbb{Q}/\langle p(x)\rangle$.

So if $p(x)$ is a polynomial over $\Bbb{Q}$, then the answer can be either $\Bbb{Q}(c)$ or $\Bbb{Q}[x]/\langle x^5+2x^3+4x^2+6\rangle$.

Correct?

Answer 1

How about $1$ in $\Bbb F_{13}$?