Field extension

Simple extension

A field extension is called simple iff is generated by the adjunction of a single element, field i.e. for some .¹ Such a is called a primitive element.

Classification

Let be a simple extension, and consider the evaluation map

Then

1. If is injective then is an infinite extension, whence is isomorphic to the field of rational functions ;
2. If is not injective then is an extension of finite degree , and

where is the minimal polynomial of .

Proof

By the First isomorphism theorem, the image of is isomorphic to . Since is an integral domain, so is , and thus by Condition for a quotient commutative ring to be an integral domain, must be a prime ideal in .

First, consider , i.e. is injective. By the universal property for the Field of fractions, extends to a unique homomorphism

where is a field containing and , whence by definition . By injectivity, are linearly independent so we have an infinite extension.

Now consider . Since is a Euclidean domain and thus a PID, it follows for a unique monic irreducible nonconstant polynomial . Since is maximal in , the image of is a subfield containing , and by the same token as above we have , giving the claimed isomorphism.

Properties

1. If and have the same minimal polynomial , then there exists a unique isomorphism of field extensions such that .
2. More generally, an isomorphism of ground fields such that lifts to an isomorphism of extensions .

Proof of 1–2

Note that an isomorphism of simple field extensions is completely determined by the image of the primitive element.

Now using the isomorphism described in Classification,

which necessarily fixes and maps , proving ^P1, whereof ^P2 is a straightforward generalization.

Results

• Bound on the automorphism group of a finite simple extension.
• Simplicity of an algebraic extension
• By the Primitive element theorem, every finite separable extension is simple.

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Footnotes

1. 2009. Algebra: Chapter 0, §VII.1.2, pp. 387–388 ↩