gives an embedding of the finite field ff1 in the finite field ff2.
represents the embedding of the ambient field of e1 in the ambient field of e2, which maps e1 to e2.
- Finite field embeddings are also known as Galois field embeddings or finite field monomorphisms.
- Finite field embeddings are typically used to identify one finite field with a subfield of another.
- If ℰ=FiniteFieldEmbedding[e1e2], where e1∈ff1 and e2∈ff2, then maps ff1 into ff2, , and for all .
- A finite field ff1 can be embedded in ff2 if it has the same characteristic as ff2 and its extension degree divides that of ff2.
- Finite field elements e1∈ff1 and e2∈ff2 define a field embedding of ff1 in ff2 iff they have the same MinimalPolynomial and e1 generates ff1. The latter condition is satisfied iff the degree of the minimal polynomial of e1 is equal to the extension degree of ff1 over .
- For an embedding ℰ=FiniteFieldEmbedding[e1e2], ℰ["Projection"] represents a linear mapping from the ambient field ff2 of e2 onto the ambient field ff1 of e1, treated as vector spaces over , such that for all .
Examplesopen allclose all
Basic Examples (1)
Use the Extension option to combine the last two steps:
Properties & Relations (4)
Use FiniteFieldElementTrace to compute :
Use FiniteFieldElementNorm to compute :
Use MinimalPolynomial to find the minimal polynomial of an element of over :
Use Composition to compose finite field embeddings:
Wolfram Research (2023), FiniteFieldEmbedding, Wolfram Language function, https://reference.wolfram.com/language/ref/FiniteFieldEmbedding.html.
Wolfram Language. 2023. "FiniteFieldEmbedding." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FiniteFieldEmbedding.html.
Wolfram Language. (2023). FiniteFieldEmbedding. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FiniteFieldEmbedding.html