Categorical Galois Theory and the Galois Correspondence

Semi-topological Galois Theory

Algebra Colloquium ◽

10.1142/s1005386715000589 ◽

2015 ◽

Vol 22 (04) ◽

pp. 687-706

Author(s):

Hsuan-Yi Liao ◽

Jyh-Haur Teh

Keyword(s):

Galois Theory ◽

Galois Groups ◽

Braid Groups ◽

Covering Spaces ◽

Cyclic Groups ◽

Inverse Galois Problem ◽

Field Extensions ◽

Galois Correspondence

We introduce splitting coverings to enhance the well known analogy between field extensions and covering spaces. Semi-topological Galois groups are defined for Weierstrass polynomials and a Galois correspondence is proved. Combining results from braid groups, we solve the topological inverse Galois problem. As an application, symmetric and cyclic groups are realized over ℚ.

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On the Galois correspondence theorem in separable Hopf Galois theory

Publicacions Matemàtiques ◽

10.5565/publmat_60116_08 ◽

2016 ◽

Vol 60 ◽

pp. 221-234 ◽

Cited By ~ 4

Author(s):

T. Crespo ◽

A. Rio ◽

M. Vela

Keyword(s):

Galois Theory ◽

Galois Correspondence ◽

Correspondence Theorem

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On the Krull Galois theory for non-algebraic extension fields

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700046724 ◽

1971 ◽

Vol 4 (3) ◽

pp. 367-387 ◽

Cited By ~ 2

Author(s):

T. Soundararajan ◽

K. Venkatachaliengar

Keyword(s):

Galois Group ◽

Galois Theory ◽

Algebraic Extension ◽

Transcendence Degree ◽

Extension Field ◽

Translation Invariant ◽

Intermediate Field ◽

Galois Correspondence

The Krull Galois theory for infinite separable normal extensions is generalized in this note to non-algebraic extensions. For any extension field E of a field K it is shown that the Galois group G can be given a translation invariant topology such that the closed subgroups are precisely the subgroups that figure in a Galois correspondence. For extension fields E/K such that E/K is of finite transcendence degree and such that E is Galois over each intermediate field the topology turns out to be compact and we have a Galois correspondence in the Krull fashion. For infinite transcendence degree extensions the Galois correspondence remains but compactness is lost. The topology coincides with the Krull topology in the case of algebraic extensions. Further properties of the topology are also studied.

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Real Liouvillian Extensions of Partial Differential Fields

Symmetry Integrability and Geometry Methods and Applications ◽

10.3842/sigma.2021.095 ◽

2021 ◽

Author(s):

Teresa Crespo ◽

◽

Zbigniew Hajto ◽

Rouzbeh Mohseni ◽

◽

...

Keyword(s):

Dynamical Systems ◽

Galois Theory ◽

Uniqueness Result ◽

Real Closed Field ◽

Closed Field ◽

Partial Differential ◽

Linear Algebraic Groups ◽

Galois Correspondence ◽

Definition Of ◽

Differential Fields

In this paper, we establish Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally p-adic differential fields with a p-adically closed field of constants. For an integrable partial differential system defined over such a field, we prove that there exists a formally real (resp. formally p-adic) Picard-Vessiot extension. Moreover, we obtain a uniqueness result for this Picard-Vessiot extension. We give an adequate definition of the Galois differential group and obtain a Galois fundamental theorem in this setting. We apply the obtained Galois correspondence to characterise formally real Liouvillian extensions of real partial differential fields with a real closed field of constants by means of split solvable linear algebraic groups. We present some examples of real dynamical systems and indicate some possibilities of further development of algebraic methods in real dynamical systems.

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Algebraic types and automorphism groups

Journal of Symbolic Logic ◽

10.2307/2275335 ◽

1993 ◽

Vol 58 (1) ◽

pp. 232-239 ◽

Cited By ~ 2

Author(s):

Akito Tsuboi

Keyword(s):

General Model ◽

Galois Extension ◽

Galois Theory ◽

Automorphism Groups ◽

The Other ◽

Closed Field ◽

Prime Characteristic ◽

Galois Correspondence ◽

Definable Functions ◽

Special Case

Galois theory states that if L is a certain algebraic extension (called a Galois extension) of a field K, then there is a one-to-one correspondence (called a Galois correspondence) between subfields M, K ⊂ M ⊂ L and subgroups of the automorphism groups of L fixing the elements in K.A subfield of a field L can be considered as a substructure of L in general model theory. However, a substructure is a subset closed under functions which are interpretations of function symbols in a given language, so the notion of substructure may change if we expand the language by adding definable notions. On the other hand a definably closed substructure is a subset which is closed under all definable functions, and it does not change by such expansions. If we are interested in subfields of an algebraically closed field of characteristic 0, these two notions are the same. But in a field of prime characteristic they are not equal. Speaking roughly, a Galois extension of a field K is an extension whose subfields are relatively definably closed. Poizat [4] showed that if a structure M has elimination of imaginaries there is a kind of Galois correspondence between definably closed substructures and subgroups of bijective elementary mappings of M.In this paper, using Poizat's result we study algebraic types. As is well known, one motivation for developing the Galois theory was to show the unsolvability of equations with degree ≥ 5. We want to take this unsolvability as a special case of general phenomena. For this purpose, we introduce several notions which are stronger than mere algebraicity and study relations between these notions and groups of bijective elementary mappings. (See Theorems 3.7 and 3.9.)

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SOME DEFINABLE GALOIS THEORY AND EXAMPLES

Bulletin of Symbolic Logic ◽

10.1017/bsl.2017.2 ◽

2017 ◽

Vol 23 (2) ◽

pp. 145-159 ◽

Cited By ~ 6

Author(s):

OMAR LEÓN SÁNCHEZ ◽

ANAND PILLAY

Keyword(s):

Differential Equations ◽

Galois Theory ◽

Automorphism Groups ◽

Differential Galois Theory ◽

Algebraic Differential Equations ◽

Galois Correspondence

AbstractWe make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the constants are not “closed” in suitable senses. We also improve the definitions and results on generalized strongly normal extensions from [Pillay, “Differential Galois theory I”, Illinois Journal of Mathematics, 42(4), 1998], using this to give a restatement of a conjecture on almost semiabelian δ-groups from [Bertrand and Pillay, “Galois theory, functional Lindemann–Weierstrass, and Manin maps”, Pacific Journal of Mathematics, 281(1), 2016].

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