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# Question on generating subfields (field extensions).

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1. Let be a field extension and be a subset we define the intermediate field as the subfield of where we "add" the elements of and denote this subfield as .

How does one figure out how the elements of such a field look like?

Example

Suppose we take the extension then we let the subset of be for example what how can we describe the elements of .

What I mean by that is or .

I believe these are called the Gaussian integers.But when the subset has a few elements more it's confusing me as to what they elements look like.

Hopefully what I've posted makes some sense at least to describe what I don't get. Any algebra legends out there?
2. For nicer field extensions (algebraic extensions) things aren't bad. Let K be a field and let f(X) be an irreducible degree d polynomial over K. Then we can build a field extension (slight abuse of `=' follows)

where X in L behaves like a root of f; so we write where is a root of f (usually in C, but really whatever the algebraic closure of K is).

To be super concrete,

What about adjoining many roots? Good news about separable extensions (which all algebraic extensions over Q are), they all have the form ! This is the primitive element theorem. For example .

When you want to add transcendental elements writing down general elements isn't so nice. For instance , the field of rational functions over Q, because e is transcendental over Q - therefore expressions cannot be simplified as e does not satisfy any polynomial over Q.

This can answer your more general question., the field of rational functions in two variables over .

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