Computing Discriminants

Abstract Algebra · Axiom Academy

EXAMPLE Computing Discriminants Using the discriminant to determine whether the Galois group is A₃ or S₃ Excellent work! You've successfully computed the discriminant and determined the Galois group. Here's what we learned: Discriminant Formula: For a reduced cubic x³ + px + q, the discriminant is Δ = -4p³ - 27q² Perfect Square Test: Check if Δ is a perfect square in the base field (here ℚ). If Δ = k² for some k ∈ ℚ, then the Galois group is A₃ A₃ vs S₃: When Δ is a perfect square, Gal(f) ⊆ A₃ (the alternating group, order 3). When Δ is not a perfect square, Gal(f) = S₃ (the symmetric group, order 6) Geometric Interpretation: The discriminant being a square means the splitting field is contained in the base field, reducing the complexity of the Galois group General Principle: The discriminant always indicates whether the Galois group lies in the alternating group—this works for polynomials of any degree! This technique is fundamental in Galois theory! You can apply this same discriminant test to determine Galois groups for other polynomials. Try computing the discriminant for x³ + 3x + 5—you'll find it's -459, which is not a perfect square, so its Galois group is S₃!

This is the written version of the interactive lesson above. See the full Abstract Algebra course.