Symmetries of Fields

Abstract Algebra · Axiom Academy

Discover which rearrangements of a field preserve its structure. Only special permutations keep everything working! Let's explore numbers of the form , where a and b are rational numbers. Click on any element to see its approximate value! A field has two operations: addition and multiplication. Let's see how they work in ℚ(√2). Let's try to create a permutation. Choose where each element maps to, then test if it preserves field operations. The Obvious Symmetry: Identity The simplest symmetry is to leave everything alone! The Surprising Symmetry: Conjugation What if we swap √2 with -√2? Let's check if this preserves field operations! Let's see why swapping different elements breaks the field structure. This breaks the field structure! The field ℚ(√2) has exactly two automorphisms : These form a group under composition, isomorphic to ℤ₂! An automorphism of a field F is a bijection σ: F → F that preserves: Addition: σ(a + b) = σ(a) + σ(b) Multiplication: σ(a · b) = σ(a) · σ(b) The set of all automorphisms forms the automorphism group Aut(F).

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