In
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 ...
, the Golod–Shafarevich theorem was proved in 1964 by
Evgeny Golod and
Igor Shafarevich. It is a result in non-commutative
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
which solves the class field tower problem, by showing that class field towers can be infinite.
The inequality
Let ''A'' = ''K''⟨''x''
1, ..., ''x''
''n''⟩ be the
free algebra
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the ...
over a
field ''K'' in ''n'' = ''d'' + 1 non-commuting variables ''x''
''i''.
Let ''J'' be the 2-sided ideal of ''A'' generated by homogeneous elements ''f''
''j'' of ''A'' of degree ''d''
''j'' with
:2 ≤ ''d''
1 ≤ ''d''
2 ≤ ...
where ''d''
''j'' tends to infinity. Let ''r''
''i'' be the number of ''d''
''j'' equal to ''i''.
Let ''B''=''A''/''J'', a
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, ...
. Let ''b''
''j'' = dim ''B''
''j''.
The ''fundamental inequality'' of Golod and Shafarevich states that
::
As a consequence:
* ''B'' is infinite-dimensional if ''r''
''i'' ≤ ''d''
2/4 for all ''i''
Applications
This result has important applications in
combinatorial group theory In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations. It is much used in geometric topology, the fundamental group of a simplicial complex having in a na ...
:
* If ''G'' is a nontrivial finite
p-group
In mathematics, specifically group theory, given a prime number ''p'', a ''p''-group is a group in which the order of every element is a power of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a nonnegative integ ...
, then ''r'' > ''d''
2/4 where ''d'' = dim ''H''
1(''G'',Z/''p''Z) and ''r'' = dim ''H''
2(''G'',Z/''p''Z) (the mod ''p''
cohomology groups of ''G''). In particular if ''G'' is a finite
p-group
In mathematics, specifically group theory, given a prime number ''p'', a ''p''-group is a group in which the order of every element is a power of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a nonnegative integ ...
with minimal number of generators ''d'' and has ''r'' relators in a given presentation, then ''r'' > ''d''
2/4.
* For each prime ''p'', there is an infinite group ''G'' generated by three elements in which each element has order a power of ''p''. The group ''G'' provides a counterexample to the
generalised Burnside conjecture: it is a
finitely generated infinite
torsion group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements.
For exam ...
, although there is no uniform bound on the order of its elements.
In
class field theory
In mathematics, 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 fields using objects associated to the ground field.
Hilbert is credit ...
, the class field tower of a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
''K'' is created by iterating the
Hilbert class field In algebraic number theory, the Hilbert class field ''E'' of a number field ''K'' is the Maximal abelian extension, maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' ...
construction. The class field tower problem asks whether this tower is always finite; attributed this question to Furtwangler, though Furtwangler said he had heard it from Schreier. Another consequence of the Golod–Shafarevich theorem is that such
towers
A tower is a tall Nonbuilding structure, structure, taller than it is wide, often by a significant factor. Towers are distinguished from guyed mast, masts by their lack of guy-wires and are therefore, along with tall buildings, self-supporting ...
may be
infinite (in other words, do not always terminate in a field equal to its
Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad ...
class field). Specifically,
* Let ''K'' be an imaginary quadratic field whose
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...
has at least 6 prime factors. Then the maximal unramified 2-extension of ''K'' has infinite degree.
More generally, a number field with sufficiently many prime factors in the discriminant has an infinite class field tower.
References
* (in
Russian
Russian(s) may refer to:
*Russians (), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries
*A citizen of Russia
*Russian language, the most widely spoken of the Slavic languages
*''The Russians'', a b ...
)
*
* (in
Russian
Russian(s) may refer to:
*Russians (), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries
*A citizen of Russia
*Russian language, the most widely spoken of the Slavic languages
*''The Russians'', a b ...
)
* See Chapter 8.
* Johnson, D.L. (1980). "Topics in the Theory of Group Presentations" (1st ed.).
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
. . See chapter VI.
*
*
*
*
Serre, J.-P. (2002), "Galois Cohomology,"
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
. . See Appendix 2. (Translation of ''Cohomologie Galoisienne'', Lecture Notes in Mathematics 5, 1973.)
{{DEFAULTSORT:Golod-Shafarevich theorem
Class field theory
Theorems in group theory