Splitting Fields

Abstract Algebra · Axiom Academy

The smallest field extension where a polynomial factors completely into linear terms 1. The Splitting Field Definition f(x) factors completely into linear factors in K[x] K is generated by the roots of f over F (minimality) In other words, K contains all the roots we need, but nothing more than what's generated by those roots. If in K[x] , then . To construct the splitting field, we follow an iterative process: Start with K_0 = F and factor f(x) as much as possible If f has an irreducible factor p(x) of degree > 1 , adjoin a root: Factor f(x) over K_1 . Repeat until all factors are linear The final field K_n is the splitting field The splitting field is unique up to isomorphism (fixing F ), though the construction process may vary depending on which irreducible factor we choose at each step. Let's find the splitting field of f(x) = x^2 - 2 over . Over : f(x) = x^2 - 2 is irreducible Both roots are in K_1 , so we're done! Now let's tackle f(x) = x^3 - 2 over . This is more interesting! The splitting field is . We need both the cube root of 2 and the cube roots of unity! Two fundamental results about splitting fields: The splitting field is where is a primitive n th root of unity If , we often need to adjoin both and The degree divides where is Euler's totient function Understanding splitting fields is crucial for Galois theory, where we study the symmetries (automorphisms) of these field extensions.

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