Field theory MOC

Splitting field

Let be a field and be a polynomial of degree . The splitting field for over is an extension such that

splits in , and . It is unique up to isomorphism with1

Proof

We construct the splitting field by iterating the process of adjoining a root to a field. Let Suppose the statement and bound have been proven for polynomials with . Let be an irreducible factor of , so that

is an extension of degree , in which has a linear factor , so letting gives so the splitting field of over exists with . It follows that is a splitting field for over and

as claimed.

Now suppose that is an isomorphism of fields, and let such that , and let be a splitting field for over . Consider the composite extension . Since is algebraic, by Embedding an algebraic extension into an algebraically closed field there exists a morphism

where . Since where are the roots of , it follows

where are the roots of in , so is independent of the chosen morphism and the splitting field .

Properties

  1. Suppose is the splitting field of , and that is a factor of . Then contains a unique subfield which is the splitting field for .
Proof of 1

Let

as above. Then for some subset of indices and some ,

Then is the splitting field of , and indeed is the only such field contained in since are the only roots of in , proving ^P1

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Footnotes

  1. 2009. Algebra: Chapter 0, §VII.4.1, pp. 429–430 ↩