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?
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?
x Turn on thread page Beta
Question on generating subfields (field extensions). watch
- Thread Starter
- 09-06-2016 00:18
- 09-06-2016 14:25
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 .Last edited by FireGarden; 09-06-2016 at 14:30.