Construction of a field

Question

Given the polynomial

$$f(x)= x^4-16x^2+4$$

which has $a=\sqrt 3+\sqrt 5$ as one of its roots in $\Bbb C$, can you use $f(x)$ to construct a field $E$ of the form $Q[x]/I $ for some appropriate ideal $I$?

And does $f(x)$ factorize over this field $E$?

Any help would be appreciated thankyou!! :)

I know that f(x) is irreducible over Z_5 if that is useful.

Answer 1

$f$ is irreducible over the rationals, so, if you let $I$ be the ideal generated by $f$, then ${\bf Q}[x]/I$ is a field $E$, and $f$ has a zero in this field. In fact, $f$ splits into linear factors over this field.