Mathematics · Abstract Algebra
Let p > 2 be a prime number. Let Fp denote the field with p elements and F̄p be an algebraic closure of Fp. Which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1The set {a + b | a, b ∈ Fp} is not contained in Fp.
- 2Fp[X] has only finitely many irreducible polynomials.
- 3Let f, g ∈ Fp[X] be irreducible polynomials of the same degree and α be a root of f in F̄p. Then Fp(α) is never the splitting field of g in F̄p.
- 4Let f(X) ∈ Fp[X] and α be a root of f in F̄p. Then Fp(α) is the splitting field of f in F̄p.✓
Correct answer
Let f(X) ∈ Fp[X] and α be a root of f in F̄p. Then Fp(α) is the splitting field of f in F̄p.
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