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!
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!