Overfield

If is a subfield of , is called an overfield of .
If are determined such that an arbitrary element of can be expressed as a linear combination of as in


, then is called a base.
If is finite, then is called an overfield of .
If is infinite, then is an infinite overfield of .
If a polynomial with root exists in with , then is algebraic.
If is irreducible in , then is called a minimal polynomial.
Consider as a mapping from the polynomial ring to the subring of .

is a surjection homomorphism mapping, and


Consider that .
If can be decomposed into first-order factors in , then is called a decomposition field of .

:Fields
:Isomorphism mapping
If , then the following expression is true:

Consider that the decomposition field of is , and the decomposition field of is .
In this case, the isomorphism mapping resulting from the extension of exists.
When all irreducible polynomials of have a single root, is called to be separative on .

[0]Top



Japanese sites
Mathematical Formulas
Kodawari House
Pinpoint StreetView
Excel VBA Techniques
Excel Formula Analysis