Abstract Algebra: Theory and Applications · Problem 17.5.20

Integral Domains — Thomas Judson

Mathematics — Abstract Algebra · cyclotomic-polynomials

Problem

Cyclotomic Polynomials. The polynomial Φp(x)=xp−1x−1=xp−1+xp−2+⋯+x+1\Phi_p(x) = \frac{x^p - 1}{x - 1} = x^{p-1} + x^{p-2} + \cdots + x + 1 for pp prime is called the cyclotomic polynomial. Show that Φp(x)\Phi_p(x) is irreducible over Q\mathbb{Q} for any prime pp. Hint. The polynomial Φn(x)=xn−1x−1=xn−1+xn−2+⋯+x+1\Phi_n(x) = \frac{x^n-1}{x-1} = x^{n-1} + x^{n-2} + \cdots + x + 1 is called the cyclotomic polynomial. Show that Φp(x)\Phi_p(x) is irreducible over Q\mathbb{Q} for any prime pp.

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