Field Theory

Determine the Splitting Field of the Polynomial $x^4+x^2+1$ over $\Q$

Field theory problems and solution in abstract algebra

Problem 92

Determine the splitting field and its degree over $\Q$ of the polynomial
\[x^4+x^2+1.\] Add to solve later

Hint.

The polynomial $x^4+x^2+1$ is not irreducible over $\Q$.

Proof.

Note that we can factor the polynomial as follows.
\begin{align*}
x^4+x^2+1&=x^4+2x^2+1-x^2=(x^2+1)^2-x^2\\
&=(x^2+x+1)(x^2-x+1).
\end{align*}

Thus the roots of the polynomial are
\[ x=\frac{\pm 1 \pm \sqrt{-3}}{2}\] by the quadratic formula.

The field $\Q(\sqrt{-3})$ contains all the roots of $x^4+x^2+1$.
Hence the splitting field is a subfield of $\Q(\sqrt{-3})$, and it is not $\Q$ since the roots are not real numbers.

Since the polynomial $x^2+3$ is irreducible over $\Q$ by Eisenstein’s criterion, the extension degree $[\Q(\sqrt{-3}):\Q]=2$.
Thus the field $\Q(\sqrt{-3})$ must be the splitting field and its degree over $\Q$ is $2$.

Add to solve later

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