subclass_of abelian extension
In ___MASK90___ , more specifically in the ___MASK19___ of modern ___MASK94___ known as Galois theory , the Galois group of a certain type of ___MASK15___ is a specific group associated with the ___MASK15___ . The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory , so named in honor of Évariste Galois who first discovered them . In ___MASK90___ , a ___MASK21___ is an expression consisting of variables ( or indeterminates ) and coefficients , that involves only the operations of addition , subtraction , multiplication , and non - negative integer exponents . An example of a ___MASK21___ of a ___MASK1___ indeterminate is . An example in three variables is . In abstract ___MASK94___ , ___MASK41___ extensions are the main object of ___MASK68___ in ___MASK41___ theory . The general idea is to start with a ___MASK79___ ___MASK41___ and construct in some manner a larger ___MASK41___ that contains the ___MASK79___ ___MASK41___ and satisfies additional properties . For instance , the set Q ( 2 ) = {" a " + " b " 2 | " a ", " b "  Q } is the smallest extension of Q that includes every real solution to the equation " x " = 2 . ___MASK94___ ( from Arabic "" al - jabr "" meaning " ___MASK88___ of broken parts ") is one of the broad parts of mathematics , together with number theory , geometry and analysis . In its most general form , algebra is the ___MASK68___ of mathematical symbols and the rules for manipulating these symbols ; it is a unifying thread of almost all of mathematics . As such , it includes everything from elementary equation solving to the ___MASK68___ of abstractions such as groups , rings , and fields . The more basic parts of algebra are called elementary algebra , the more abstract parts are called abstract algebra or modern algebra . Elementary algebra is generally considered to be essential for any ___MASK68___ of mathematics , science , or engineering , as well as such applications as medicine and economics . Abstract algebra is a major area in advanced mathematics , studied primarily by professional mathematicians . In ___MASK90___ , the fundamental ___MASK46___ of Galois theory is a result that describes the structure of certain types of field extensions . In ___MASK94___ , which is a broad division of mathematics , abstract ___MASK94___ ( occasionally called modern ___MASK94___ ) is the ___MASK68___ of algebraic structures . Algebraic structures include groups , rings , fields , modules , vector spaces , lattices , and algebras . The term " abstract ___MASK94___ " was coined in the early 20th century to distinguish this area of ___MASK68___ from the other parts of ___MASK94___ . In ___MASK90___ , and more specifically in abstract ___MASK94___ , an algebraic structure is a set ( called carrier set or underlying set ) with one or more finitary operations defined on it that satisfies a list of axioms . In abstract ___MASK94___ , an abelian extension is a Galois extension whose Galois group is abelian . When the Galois group is also cyclic , the extension is also called a cyclic extension . Going in the other direction , a Galois extension is called solvable if its Galois group is solvable , i . e . , if the group can be decomposed into a series of normal extensions of an abelian group . Every finite extension of a finite ___MASK41___ is a cyclic extension . Class ___MASK6___ provides detailed information about the abelian extensions of number fields , function fields of algebraic curves over finite fields , and local fields . There are two slightly different definitions of the term cyclotomic extension . It can mean either an extension formed by adjoining roots of unity to a ___MASK41___ , or a subextension of such an extension . The cyclotomic fields are examples . A cyclotomic extension , under either definition , is always abelian . If a ___MASK41___ K contains a primitive n - th root of unity and the n - th root of an element of K is adjoined , the resulting so - called Kummer extension is an abelian extension ( if K has characteristic p we should say that p does n ' t divide n , since otherwise this can fail even to be a separable extension ) . In general , however , the Galois groups of n - th roots of elements operate both on the n - th roots and on the roots of unity , giving a non - abelian Galois group as semi - direct product . The Kummer theory gives a complete description of the abelian extension case , and the Kronecker -- Weber ___MASK46___ tells us that if K is the ___MASK41___ of rational numbers , an extension is abelian if and only if it is a subfield of a ___MASK41___ obtained by adjoining a root of unity . There is an important analogy with the fundamental group in topology , which classifies all covering spaces of a space : abelian covers are classified by its abelianisation which relates directly to the first homology group . In ___MASK90___ , an automorphism is an ___MASK96___ from a mathematical object to itself . It is , in some sense , a ___MASK12___ of the object , and a way of mapping the object to itself while preserving all of its structure . The set of all automorphisms of an object forms a group , called the automorphism group . It is , loosely speaking , the ___MASK12___ group of the object . ___MASK90___ ( from ___MASK45___  " máthma ",  knowledge , ___MASK68___ , learning ) is the ___MASK68___ of topics such as quantity ( numbers ), structure , space , and change . There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics . Évariste Galois ( 25 October 1811  31 May 1832 ) was a ___MASK60___ ___MASK32___ born in Bourg - la - Reine . While still in his teens , he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals , thereby solving a ___MASK71___ standing for 350 years . His work laid the foundations for Galois theory and group theory , two major branches of abstract algebra , and the subfield of Galois connections . He died at ___MASK93___ 20 from wounds suffered in a duel . In ___MASK90___ , a Galois extension is an algebraic ___MASK15___ " E "/" F " that is normal and separable ; or equivalently , " E "/" F " is algebraic , and the field fixed by the automorphism group Aut (" E "/" F ") is precisely the base field " F ". The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory . In ___MASK90___ , more specifically in abstract ___MASK94___ , Galois theory , named after Évariste Galois , provides a ___MASK89___ between ___MASK6___ and group theory . Using Galois theory , certain problems in ___MASK6___ can be reduced to group theory , which is , in some sense , simpler and better understood .