Factoring in Polynomial Quotient Ring

[Chris*]

I'm really stuck on this problem.

Question: Let $\displaystyle p=x^4-2$. The factor $\displaystyle K=\mathbb{Q}[x]/(p)$ is a field since $\displaystyle p$ is irreducible over $\displaystyle \mathbb{Q}$.

Factor the polynomial $\displaystyle q=y^4-2$ in $\displaystyle K$ into a product of irreducible polynomials, and show that each factor is irreducible.

Solution Attempt:
So, I know that $\displaystyle K$ is an extension that does have a root of $\displaystyle p$, namely $\displaystyle x$ itself.
So $\displaystyle y-x$ must be a factor of $\displaystyle q$.
Replacing $\displaystyle 2$ with $\displaystyle x^4$ in $\displaystyle q$, I got:
$\displaystyle q=y^4-2=y^4-x^4=(y-x)(y+x)(y^2+x^2)$

Now, I know that $\displaystyle (y-x),(y+x)$ are irreducible because they are of degree one. But I'm not sure how to show that $\displaystyle (y^2+x^2)$ is irreducible in K.
Since it's a quadratic polynomial, I wanted to show its roots are not in K, but I wasn't sure how to go about doing that.

I appreciate any suggestions!

 

[daon2]

I'm really stuck on this problem.

Question: Let $\displaystyle p=x^4-2$. The factor $\displaystyle K=\mathbb{Q}[x]/(p)$ is a field since $\displaystyle p$ is irreducible over $\displaystyle \mathbb{Q}$.

Factor the polynomial $\displaystyle q=y^4-2$ in $\displaystyle K$ into a product of irreducible polynomials, and show that each factor is irreducible.

Solution Attempt:
So, I know that $\displaystyle K$ is an extension that does have a root of $\displaystyle p$, namely $\displaystyle x$ itself.
So $\displaystyle y-x$ must be a factor of $\displaystyle q$.
Replacing $\displaystyle 2$ with $\displaystyle x^4$ in $\displaystyle q$, I got:
$\displaystyle q=y^4-2=y^4-x^4=(y-x)(y+x)(y^2+x^2)$

Now, I know that $\displaystyle (y-x),(y+x)$ are irreducible because they are of degree one. But I'm not sure how to show that $\displaystyle (y^2+x^2)$ is irreducible in K.
Since it's a quadratic polynomial, I wanted to show its roots are not in K, but I wasn't sure how to go about doing that.

I appreciate any suggestions!

Can you show $\displaystyle K[y]/(y^2+x^2)$ is a field? If so, the ideal must be maximal, and hence the polynomial irreducible. Any element in this ring can be written $\displaystyle ay+b$ where $\displaystyle a,b \in K = Q[x]/(p)$. Really all you need is multiplicative inverses, it might help to treat the "polynomials" in $\displaystyle K$ as just real numbers since it is a field afterall.