# Construction of a field

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.

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Dear user49799, Welcome to math.SE. since you are a new user, we wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say what your thoughts on the problem are so far; this will prevent people from telling you things you already know, and help them write their answers at an appropriate level. Further, it would be better if you could typeset your problem so that it is easy for people to read. Kindly look here meta.math.stackexchange.com/questions/5020/… for more details on typesetting. – user17762 Nov 17 '12 at 21:07
thanks! i will do that from now on. – user49799 Nov 17 '12 at 21:13

## 1 Answer

$f$ 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.

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