Abstract Algebra II · Polynomial rings and irreducibility
Reciprocal polynomials preserve irreducibility
Problem
Let $F$ be a field and let $f=a_0+a_1x+\cdots+a_nx^n\in F[x]$ with $a_0a_n\ne0$. Define $f^*(x)=x^nf(x^{-1})$. Prove that $f$ is irreducible if and only if $f^*$ is irreducible.
Hint
The hypotheses ensure $\deg f^*=n$ and $(f^*)^*=f$.
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