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Problem 1

Avg difficulty 3.0

Determine which of the following polynomials, if any, are irreducible over Q\mathbb{Q}.

  1. f1(x)=x3+x2+x+1f_{1}(x)=x^{3}+x^{2}+x+1

  2. f2(x)=6x3+x2+x+1f_{2}(x)=6x^{3}+x^{2}+x+1

  3. f3(x)=x3+9f_{3}(x)=x^{3}+9

  4. f4(x)=x4+5f_{4}(x)=x^{4}+5

  5. f5(x)=x4−2x3+x2−x+1f_{5}(x)=x^{4}-2x^{3}+x^{2}-x+1

  6. f6(x)=x4+12x3+54x2+108x+86f_{6}(x)=x^{4}+12x^{3}+54x^{2}+108x+86

Kan, A Complete Introduction to Abstract Algebra, Rings, Factorization of Polynomials, Problem 1

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Adapted from A Complete Introduction to Abstract Algebra © 2022–2026 Kan Onn Kit. Licensed under CC BY-SA 4.0. Source: A Complete Introduction to Abstract Algebra. Imported at revision a559e8b.