Common Polynomial Irreducibility Criteria

Explanations 4

• Perron's Criterion Let f(x)=xn+an−1xn−1+...+a1x+a0f(x) = x^n + a_{n - 1}x^{n - 1} + ... + a_1x + a_0f(x)=xn+an−1​xn−1+...+a1​x+a0​ be a polynomial with integer coefficients with…

  By Zi Song Yeoh

• Note: This explanation aims not only to prove common irreducibility criteria but to also demonstrate common methods used in the proofs which appear elsewhere. There are several …

  By Vincent Huang

• Prove that xp−1+xp−2+…+1x^{p-1}+x^{p-2}+\ldots+1xp−1+xp−2+…+1 is irreducible for a prime ppp. Hint: Replace xxx by x+1x+1x+1.

  By amaterasu

• Let f(x)=xn+5xn−1+3f(x) = x^n + 5x^{n-1} + 3f(x)=xn+5xn−1+3, where n>1n > 1n>1 is an integer. Prove that f(x)f(x)f(x) cannot be expressed as the product of two non-constant poly…

  By amaterasu