Mathematics · Field Theory
Let p > 2 be a prime number. Let &Double_struck;Fp denote the finite field with p elements and ‾&Double_struck;Fp be its algebraic closure. Which of the following statements is FALSE?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1The set {a + b | a, b ∈ &Double_struck;Fp} is equal to the set {a2 + b2 | a, b ∈ &Double_struck;Fp}.
- 2&Double_struck;Fp[X] has infinitely many irreducible polynomials.
- 3For any irreducible polynomials f, g ∈ &Double_struck;Fp[X] of the same degree, if α is a root of f in ‾&Double_struck;Fp, then &Double_struck;Fp(α) contains a root of g but is not the splitting field of g.✓
- 4For any f(X) ∈ &Double_struck;Fp[X] and a root α of f in ‾&Double_struck;Fp, &Double_struck;Fp(α) is the splitting field of f in ‾&Double_struck;Fp.
Correct answer
For any irreducible polynomials f, g ∈ &Double_struck;Fp[X] of the same degree, if α is a root of f in ‾&Double_struck;Fp, then &Double_struck;Fp(α) contains a root of g but is not the splitting field of g.
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