Lecture 06 - Jan 28

Recap

Field Extensions

Let

Then by the Tower Law,

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Field Generated by a Set

If is finite, then

If is not finite, then

Moreover,

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Existence of a Root

If

then there exists a field extension such that contains a root of .

If are two roots of an irreducible polynomial over , then

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Simple and Finite Extensions

If is finite, then is a finitely generated extension.

If is finite, then every element of is algebraic over .

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Algebraic Elements

Let and .

Then is algebraic over if there exists

such that

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Minimal Polynomial

If is algebraic over , then the minimal polynomial

is the monic polynomial of least degree in such that

If

and , then the minimal polynomials over and satisfy

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Composite Field Extension

Let and be field extensions.

The composite field is the smallest field containing both and .

If and are algebraic (or finite), then

is algebraic.

Moreover,

and the right inequality can be strict.

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-Algebra

Let be a commutative ring with unity.

An -algebra is a ring together with a ring homomorphism

This induces an -module structure on via

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-Algebra Homomorphism

Let and be -algebras.

An -algebra homomorphism

is a ring homomorphism such that the diagram commutes, i.e.,

Equivalently,

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Automorphism Groups

Let be a field extension.

Define:

• : the set of all field automorphisms of .
• : the set of all -algebra automorphisms of .

If , then for all ,

Thus,

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Special Cases

If , then

If , then