I've been struggling with factoring polynomials into irreducibles. I found an example and would like to know how to tackle it. If our polynomial is $x^4+1$, what are the irreducible polynomial factors in $\mathbb{Q}[x]$? What about in $\mathbb{F}_2[x]$?
Factoring $x^4+1$ into irreducibles over $\mathbb{Q}$ and over $\mathbb{F}_2$
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1For $\mathbb{F}_2[x]$, maybe factorization is obvious. Or the slow way, note that $1$ is a root. So $x+1$ divides $x^4+1$. Do the division. See what happens. Remember that in general $-a=a$. – 2011-11-30