Can someone please help me to prove this:
Prove that $1+x+x^2+x^3+x^4+x^5+x^6$ is irreducible in Q[X]
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As a hint, prove that the evaluation $x\mapsto x+a$ preserves irreducibility. Then with a careful choice of $a$, use Eisenstein's criterion. – Ignacio Rojas Dec 04 '17 at 05:01
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This is a cyclotomic polynomial. – Angina Seng Dec 04 '17 at 05:19
1 Answers
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Note that "horizontally translating" a polynomial in $R[t]$ does not change it's zeros that is, replacing $x$ with $x+k$ for some $k\in R$ where $R$ is a commutative ring.
Baring that in mind, we have
$1+(x+1)+(x+1)^2+(x+1)^3+(x+1)^4+(x+1)^5+(x+1)^6$
is equal to
$x^6+7x^5+21x^4+35x^3+35x^2+21x+7$ of which Eisenstein's critereon applies to taking $q=7$.
Andrew Tawfeek
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