field extension

A field extension is a special case of ring extension.

In particular, this applies to finite field extensions of K.

One can then view morphisms in Field as field extensions.

The number of algebraically independent transcendental elements in a field extension.

Alternatively, constructing such field extensions can also be done, if a bigger container is already given.

A field extension generated by the complete factorisation of a polynomial.

The Target Field extension in 2009 added a few more tenths to the length.

Also in this case, twisted curves are isomorphic over the field extension given by the twist degree.

But this is a cyclic field extension, and so must contain a primitive root of unity.

Suppose there is some field extension L of k such that is a domain.

Given a field extension, one can "extend scalars" on associated algebraic objects.

Minimal polynomials are useful for constructing and analyzing field extensions.

Generalizing the previous example, quotient rings are often used to construct field extensions.

Let be a finitely generated field extension of a field .

For non-algebraic field extensions transcendence degree is likewise used.

Every field extension has a transcendence basis.

We can factor a polynomial , where is a finite field extension of .

Let be an arbitrary field extension.

In abstract algebra, field extensions are the main object of study in field theory.

See for example field extension.

In more advanced mathematics they play an important role in ring theory, especially in the construction of field extensions.

Thus, if 'D' is connected, the meromorphic functions form a field (mathematics), in fact a field extension of the complex numbers.

Let be a field extension of Define by if is algebraic over Then is a dependence relation.

Here, a -module V is absolutely simple if is simple for any field extension .

Now, if F is a field extension of k, then the base extension induces .