Factoring in Polynomial Quotient Ring

Chris*

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Jan 9, 2007
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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!
 
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.
 
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