The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions.
42.
Thus for testing if a polynomial is square-free, it is not necessary to consider explicitly any field extension nor to compute the roots.
43.
The extended Euclidean algorithm is the basic tool for computing multiplicative inverses in modular structures, typically the modular integers and the algebraic field extensions.
44.
This generalizes complex conjugation and the conjugation of degree 2 finite field extensions, and allows one to define Hermitian forms and unitary groups as above.
45.
It is easy to construct field extensions with any given finite group as Galois group, as long as one does not also specify the ground field.
46.
If E \ supseteq L \ supseteq F are field extensions, then is separable over if and only if is separable over, and is separable over.
47.
Let K ( \ alpha ) be an algebraic field extension generated by an element \ alpha, which has P ( x ) as minimal polynomial.
48.
The author is not trying to be fancy or trying to impress anyone by using the technical terms " abelian " and " field extension ".
49.
Respectively, the tools of a compass and straightedge allow us to create circles degree of the field extension corresponding to each new coordinate is 2 or 1.
50.
In case of E = L, the definition above is consistent with the results about the adele ring in case of a field extension L / K.
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