Grothendieck's Galois Theory
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Grothendieck's Galois theory is an abstract approach to the
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
of fields, developed around 1960 to provide a way to study the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
of
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
in the setting of
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
. It provides, in the classical setting of field theory, an alternative perspective to that of
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...
based on
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
, which became standard from about the 1930s. The approach of
Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
is concerned with the category-theoretic properties that characterise the categories of finite ''G''-sets for a fixed profinite group ''G''. For example, ''G'' might be the group denoted \hat (see profinite integer), which is the
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ca ...
of the cyclic additive groups \Z/n\Z — or equivalently the completion of the infinite cyclic group \Z for the topology of subgroups of finite
index Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...
. A finite ''G''-set is then a finite set ''X'' on which ''G'' acts through a quotient finite cyclic group, so that it is specified by giving some permutation of ''X''. In the above example, a connection with classical
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
can be seen by regarding \hat as the profinite Galois group Gal(''F''/''F'') of the
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ...
''F'' of any
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
''F'', over ''F''. That is, the automorphisms of ''F'' fixing ''F'' are described by the inverse limit, as we take larger and larger finite
splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a polyn ...
s over ''F''. The connection with geometry can be seen when we look at
covering space In topology, a covering or covering projection is a continuous function, map between topological spaces that, intuitively, Local property, locally acts like a Projection (mathematics), projection of multiple copies of a space onto itself. In par ...
s of the
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...
in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
with the origin removed: the finite covering realised by the ''z''''n'' map of the disk, thought of by means of a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
variable ''z'', corresponds to the subgroup n \Z of the fundamental group of the punctured disk. The theory of Grothendieck, published in SGA1, shows how to reconstruct the category of ''G''-sets from a ''fibre functor'' \Phi, which in the geometric setting takes the fibre of a covering above a fixed base point (as a set). In fact there is an isomorphism proved of the type :G \cong \operatorname(\Phi), the latter being the group of automorphisms (self- natural equivalences) of \Phi. An abstract classification of categories with a functor to the category of sets is given, by means of which one can recognise categories of ''G''-sets for ''G'' profinite. To see how this applies to the case of fields, one has to study the
tensor product of fields In mathematics, the tensor product of two field (mathematics), fields is their tensor product of algebras, tensor product as algebra over a field, algebras over a common subfield (mathematics), subfield. If no subfield is explicitly specified, t ...
. In
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notio ...
theory this is a part of the study of '' atomic toposes''.


See also

* Tannakian formalism * Fiber functor *
Anabelian geometry Anabelian geometry is a theory in number theory which describes the way in which the algebraic fundamental group ''G'' of a certain arithmetic variety ''X'', or some related geometric object, can help to recover ''X''. The first results for nu ...


References

* * * (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois
groupoids In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fun ...
.) * * *{{cite journal, authorlink1=Olivia Caramello , last= Caramello, first= Olivia , date=2016 , title=Topological Galois theory , journal=
Advances in Mathematics ''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed ...
, volume= 291 , pages= 646–695, doi=10.1016/j.aim.2015.11.050 , doi-access= free , arxiv=1301.0300 Galois theory Algebraic geometry Category theory