Field Embedding

A field embedding of a field K into a field L is an injective field homomorphism sigma:K->L. In particular, it preserves addition, multiplication, and the multiplicative identity:

A field embedding sigma:K->L is therefore sometimes simply defined as a nonzero field homomorphism from K to L. Here, nonzero means not the zero homomorphism, i.e., not the map sending every element of K to 0 in L. Every nonzero field homomorphism is injective since its kernel is an ideal in K and the only ideals of a field K are {0} and K.

If K/F and L/F are extension fields, an embedding of K into L over F is a field embedding sigma:K->L such that sigma(a)=a for every a in F.

For a number field, embeddings into C are classified as real embeddings or imaginary embeddings. A number field with no real embeddings is called a totally imaginary field.

See also

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References

Cite this as:

Weisstein, Eric W. "Field Embedding." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FieldEmbedding.html

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