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...

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

Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory...

In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection...

In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently...

In mathematics, Grothendieck's Galois theory is an abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the...

differential Galois theory, a variant of Galois theory dealing with linear differential equations. Galois theory studies algebraic extensions of a field...

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any...

Artin–Schreier theory is a branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic...

concerning jumps of the upper numbering filtration of the Galois group of a finite Galois extension. A special case of it when the residue fields are finite...

indeed Galois theory shows that this analogy is more than just a coincidence. The formula holds for both finite and infinite degree extensions. In the...

In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it...

This improved statement follows directly from Galois theory § A non-solvable quintic example. Galois theory implies also that x 5 − x − 1 = 0 {\displaystyle...

differential field extension generated by the solutions of a linear differential equation, using the differential Galois group of the field extension. A major goal...

differential Galois theory, but this is not strictly true. The theorem can be proved without any use of Galois theory. Furthermore, the Galois group of a...

usual Galois correspondence for subfields of a Galois extension, and Jacobson's Galois correspondence for subfields of a purely inseparable extension of...

of integers modulo n and the Galois group of Q ( ω ) . {\displaystyle \mathbb {Q} (\omega ).} This shows that this Galois group is abelian, and implies...

"Teichmüller's Lego-game and the Galois group of Q over Q" ("Un jeu de “Lego-Teichmüller” et le groupe de Galois de Q sur Q"). 3. Number fields associated...

finite Galois extension of global fields created by packaging together the various Artin L-functions associated with the extension. Each extension has many...

about Galois representations of elliptic curves. He then uses this result to prove that all semistable curves are modular, by proving that the Galois representations...

is given the name, stating that if L/K is a finite Galois extension of fields with arbitrary Galois group G = Gal(L/K), then the first cohomology group...

field extension [ F ( x ) : F ( y ) ] {\displaystyle [\mathbb {F} (x):\mathbb {F} (y)]} . This extension is generally not Galois but has Galois closure...

of an Artin L-function. Suppose that L is a finite Galois extension of the local field K, with Galois group G. If χ {\displaystyle \chi } is a character...

and attached manuscripts, Galois also constructed the general linear group over a prime field, GL(ν, p), in studying the Galois group of the general equation...

arithmetic dynamics, an arboreal Galois representation is a continuous group homomorphism between the absolute Galois group of a field and the automorphism...

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

vanishes. This is true for all finite Galois extensions of number fields, not just cyclic ones. For cyclic extensions the group H2(L/K) is isomorphic to...

inverse Galois problem for G {\displaystyle G} asks whether there exists a finite Galois extension F ∣ Q {\displaystyle F\mid \mathbb {Q} } whose Galois group...

There also exists "finite level" modifications of the Galois groups: if E/F is a finite extension, then the relative Weil group of E/F is WE/F = WF/W c...

Galois closure with Galois group Gal(K/Q) isomorphic to the cyclic group of order three. However, any other cubic field K is a non-Galois extension of...