Since the polynomial $p=x^4−2$ is irreducible over $\mathbb{Q}$, the factor ring $K=\mathbb{Q}[x]/(p)$ is a field.
I'd like to factor the polynomial $q=y^4−2$ in $K[y]$ into a product of irreducible polynomials, so that I can prove all factors in this decomposition are indeed irreducible. How should I factor the polynomial out?
Best Answer
In $K$, $x$ is a root of $p$, so $y-x$ is a factor of $q$. $y+x$ is also a factor, So $y^2-x^2$ is a factor. Now divide $q$ by $y^2-x^2$ to get the remaining factor (not forgetting to use $x^4=2$ somewhere along the way). You should get $$q=(y-x)(y+x)(y^2+x^2)$$