In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to...

theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler...

significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin...

Fundamental theorem of Galois theory Fundamental theorem of geometric calculus Fundamental theorem on homomorphisms Fundamental theorem of ideal theory in number...

One of the important structure theorems from Galois theory comes from the fundamental theorem of Galois theory. This states that given a finite Galois extension...

find applications in various mathematical theories. They generalize the fundamental theorem of Galois theory about the correspondence between subgroups...

The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial...

together with Ribet's theorem, would also prove Fermat's Last Theorem. In mathematical terms, Ribet's theorem showed that if the Galois representation associated...

solutions of polynomial equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between...

fields. Galois then used this theorem heavily in his development of the Galois group. Since then it has been used in the development of Galois theory and...

zero to non-zero characteristic. For example, the fundamental theorem of Galois theory is a theorem about normal extensions, which remains true in non-zero...

interest. The proof of the Abel–Ruffini theorem predates Galois theory. However, Galois theory allows a better understanding of the subject, and modern...

do) Angle of parallelism Galois group Fundamental theorem of Galois theory (to do) Gödel number Gödel's incompleteness theorem Group (mathematics) Halting...

subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and...

content of the fundamental theorem of Galois theory. Given a poset P = (X, ≤) (short for partially ordered set; i.e., a set that has a notion of ordering...

class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global...

number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal...

straightedge. Galois theory, devoted to understanding the symmetries of field extensions, provides an elegant proof of the Abel-Ruffini theorem that general...

Liouville's theorem is sometimes presented as a theorem in differential Galois theory, but this is not strictly true. The theorem can be proved without...

of the automorphism group; see Fundamental theorem of Galois theory. Along with a module of covariants, the ring of invariants is a central object of...

group the Galois group of a Galois extension of the rational numbers? (more unsolved problems in mathematics) In Galois theory, the inverse Galois problem...

number theory. Class field theory accomplishes this goal when K is an abelian extension of Q (that is, a Galois extension with abelian Galois group)....

extensions formed as the splitting field of a polynomial. This theory establishes—via the fundamental theorem of Galois theory—a precise relationship between fields...

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b...

topology Discrete space Fundamental group Geometry Homology Minkowski's theorem Topological group Field Finite field Galois theory Grothendieck group Group...

between the two types of objects. One may view other theorems in the same light. For example, the fundamental theorem of Galois theory asserts that there...

structure as in the proofs now given of the fundamental theorem of Galois theory, though much more complex). One of the two inequalities involved an argument...

permutation of the n roots among themselves. The significance of the Galois group derives from the fundamental theorem of Galois theory, which proves...

mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary...

coefficients, but this follows either from the fundamental theorem of Galois theory, or from the fundamental theorem of symmetric polynomials and Vieta's formulas...