Computing Gal(Q(√2, √3)/Q)

Abstract Algebra · Axiom Academy

Finding all automorphisms and identifying the Klein four-group structure Excellent work! You've successfully computed the Galois group and analyzed its structure. Here's what we learned: Automorphisms are determined by images of generators: Since Q(√2, √3) is generated by √2 and √3 over Q, any automorphism is completely determined by where it sends these two elements. Preservation constraints: Because σ(√2)² = σ(2) = 2, we have σ(√2) = ±√2. Similarly σ(√3) = ±√3. This gives us exactly 4 possibilities. Klein four-group structure: The group is abelian with every non-identity element having order 2, making it isomorphic to ℤ₂ × ℤ₂. Subgroup correspondence: The three subgroups of order 2 correspond to the three intermediate fields: Q(√2), Q(√3), and Q(√6) by the Fundamental Theorem of Galois Theory. Lattice structure: The subgroup lattice forms a diamond shape, characteristic of the Klein four-group. This example illustrates key concepts in Galois theory: how automorphisms are determined by their action on field generators, how to compute group operations, and how to identify familiar group structures. The Klein four-group appears frequently in Galois theory!

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