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In mathematics, a Witt group of a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, named after
Ernst Witt Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time. Biography Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the ...
, is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
whose elements are represented by
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear ...
s over the field.


Definition

Fix a field ''k'' of characteristic not equal to two. All
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
s will be assumed to be finite-
dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
. We say that two spaces equipped with
symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear ...
s are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector.Milnor & Husemoller (1973) p. 14 Each class is represented by the core form of a Witt decomposition.Lorenz (2008) p. 30 The Witt group of ''k'' is the abelian group ''W''(''k'') of
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of non-degenerate symmetric bilinear forms, with the group operation corresponding to the
orthogonal direct sum In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, ma ...
of forms. It is additively generated by the classes of one-dimensional forms.Milnor & Husemoller (1973) p. 65 Although classes may contain spaces of different dimension, the parity of the dimension is constant across a class and so rk : ''W''(''k'') → Z/2Z is a
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same ...
. The elements of finite order in the Witt group have order a power of 2;Lorenz (2008) p. 37Milnor & Husemoller (1973) p. 72 the
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
is the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of the
functorial In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
map from ''W''(''k'') to ''W''(''k''py), where ''k''py is the
Pythagorean closure In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field F is an extension obtained by adjoining an element \sqrt for some \lamb ...
of ''k'';Lam (2005) p. 260 it is generated by the
Pfister form In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field ''F'' of characteristic not 2. For a natural number ''n'', an ''n''-fold P ...
s \langle\!\langle w \rangle\!\rangle = \langle 1, -w \rangle with w a non-zero sum of squares.Lam (2005) p. 395 If ''k'' is not
formally real In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field. Alternative definitions The definition given above i ...
, then the Witt group is
torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
, with
exponent Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to r ...
a power of 2. The height of the field ''k'' is the exponent of the torsion in the Witt group, if this is finite, or ∞ otherwise.Lam (2005) p. 395


Ring structure

The Witt group of ''k'' can be given a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
structure, by using the
tensor product of quadratic forms In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as '' quadratic spaces''. If ''R'' is a commutative ring where 2 is invertible (that is, ''R'' has characteristic \text(R) \neq 2) ...
to define the ring product. This is sometimes called the Witt ring ''W''(''k''), though the term "Witt ring" is often also used for a completely different ring of
Witt vector In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field of orde ...
s. To discuss the structure of this ring we assume that ''k'' is of characteristic not equal to 2, so that we may identify symmetric bilinear forms and quadratic forms. The kernel of the rank mod 2 homomorphism is a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
, ''I'', of the Witt ringMilnor & Husemoller (1973) p. 66 termed the ''fundamental ideal''. The
ring homomorphisms Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film an ...
from ''W''(''k'') to Z correspond to the field orderings of ''k'', by taking
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
with respective to the ordering.Lorenz (2008) p. 31 The Witt ring is a
Jacobson ring In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every ...
.Lorenz (2008) p. 35 It is a
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noeth ...
if and only if there are finitely many
square class In mathematics, specifically abstract algebra, a square class of a field F is an element of the square class group, the quotient group F^\times/ F^ of the multiplicative group of nonzero elements in the field modulo the square elements of the fiel ...
es; that is, if the squares in ''k'' form a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
of finite
index Index (or its plural form 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 a Halo megastru ...
in the multiplicative group of ''k''.Lam (2005) p. 32 If ''k'' is not formally real, the fundamental ideal is the only prime ideal of ''W''Lorenz (2008) p. 33 and consists precisely of the
nilpotent element In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
s; ''W'' is a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
and has
Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally th ...
0.Lam (2005) p. 280 If ''k'' is real, then the nilpotent elements are precisely those of finite additive order, and these in turn are the forms all of whose signatures are zero;Lorenz (2008) p. 36 ''W'' has Krull dimension 1. If ''k'' is a real
Pythagorean field In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field F is an extension obtained by adjoining an element \sqrt for some \lambd ...
then the
zero-divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
s of ''W'' are the elements for which some signature is zero; otherwise, the zero-divisors are exactly the fundamental ideal.Lam (2005) p. 282 If ''k'' is an ordered field with positive cone ''P'' then
Sylvester's law of inertia Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if ''A'' is the symmetric matrix that defines the quadrati ...
holds for quadratic forms over ''k'' and the ''signature'' defines a ring homomorphism from ''W''(''k'') to Z, with kernel a prime ideal ''K''''P''. These prime ideals are in
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
with the orderings ''Xk'' of ''k'' and constitute the minimal prime ideal
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
MinSpec ''W''(''k'') of ''W''(''k''). The bijection is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
between MinSpec ''W''(''k'') with the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
and the set of orderings ''X''''k'' with the Harrison topology. The ''n''-th power of the fundamental ideal is additively generated by the ''n''-fold
Pfister form In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field ''F'' of characteristic not 2. For a natural number ''n'', an ''n''-fold P ...
s.Lam (2005) p.316


Examples

* The Witt ring of C, and indeed any
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
or
quadratically closed field In mathematics, a quadratically closed field is a field in which every element has a square root.Lam (2005) p. 33Rajwade (1993) p. 230 Examples * The field of complex numbers is quadratically closed; more generally, any algebraically clos ...
, is Z/2Z.Lam (2005) p. 34 * The Witt ring of R is Z. * The Witt ring of a
finite 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 field, a finite field is a set on which the operations of multiplication, addition, subtr ...
F''q'' with ''q'' odd is Z/4Z if ''q'' ≡ 3 mod 4 and
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
(Z/2Z) 'F*''/''F*''2if ''q'' ≡ 1 mod 4.Lam (2005) p.37 * The Witt ring of a
local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
with
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
of
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
congruent to 1 modulo 4 is isomorphic to the group ring (Z/2Z) 'V''where ''V'' is the Klein 4-group.Lam (2005) p.152 * The Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 is (Z/4Z) 'C''2where ''C''2 is a cyclic group of order 2.Lam (2005) p.152 * The Witt ring of Q2 is of order 32 and is given byLam (2005) p.166 ::\mathbf_8 ,t\langle 2s,2t,s^2,t^2,st-4 \rangle .


Invariants

Certain invariants of a quadratic form can be regarded as functions on Witt classes. We have seen that dimension mod 2 is a function on classes: the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
is also well-defined. The
Hasse invariant of a quadratic form In mathematics, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form ''Q'' over a field ''K'' takes values in the Brauer group Br(''K''). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt. The quadratic form ''Q'' ma ...
is again a well-defined function on Witt classes with values in the
Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...
of the field of definition.Lam (2005) p.119


Rank and discriminant

We define a ring over ''K'', ''Q''(''K''), as a set of pairs (''d'', ''e'') with ''d'' in ''K*''/''K*''2 and ''e'' in Z/2Z. Addition and multiplication are defined by: :(d_1,e_1) + (d_2,e_2) = ((-1)^d_1d_2, e_1+e_2) :(d_1,e_1) \cdot (d_2,e_2) = (d_1^d_2^, e_1e_2). Then there is a
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
ring homomorphism from ''W''(''K'') to this obtained by mapping a class to discriminant and rank mod 2. The kernel is ''I''2.Conner & Perlis (1984) p.12 The elements of ''Q'' may be regarded as classifying graded quadratic extensions of ''K''.Lam (2005) p.113


Brauer–Wall group

The triple of discriminant, rank mod 2 and Hasse invariant defines a map from ''W''(''K'') to the
Brauer–Wall group In mathematics, the Brauer–Wall group or super Brauer group or graded Brauer group for a field ''F'' is a group BW(''F'') classifying finite-dimensional graded central division algebras over the field. It was first defined by as a generalizatio ...
BW(''K'').Lam (2005) p.117


Witt ring of a local field

Let ''K'' be a complete
local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
with valuation ''v'', uniformiser π and residue field ''k'' of characteristic not equal to 2. There is an injection ''W''(''k'') → ''W''(''K'') which lifts the diagonal form ⟨''a''1,...''a''''n''⟩ to ⟨''u''1,...''u''''n''⟩ where ''u''''i'' is a unit of ''K'' with image ''a''''i'' in ''k''. This yields : W(K) = W(k) \oplus \langle \pi \rangle \cdot W(k) identifying ''W''(''k'') with its image in ''W''(''K'').Garibaldi, Merkurjev & Serre (2003) p.64


Witt ring of a number field

Let ''K'' be 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 ...
. For quadratic forms over ''K'', there is a Hasse invariant ±1 for every
finite place Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
corresponding to the
Hilbert symbol In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of reals or p-adic numbers . It is related to reciprocity ...
s. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
s coming from real embeddings.Conner & Perlis (1984) p.16 We define the symbol ring over ''K'', Sym(''K''), as a set of triples (''d'', ''e'', ''f'' ) with ''d'' in ''K*''/''K*''2, ''e'' in ''Z''/2 and ''f'' a sequence of elements ±1 indexed by the places of ''K'', subject to the condition that all but finitely many terms of ''f'' are +1, that the value on acomplex places is +1 and that the product of all the terms in ''f'' in +1. Let 'a'', ''b''be the sequence of Hilbert symbols: it satisfies the conditions on ''f'' just stated.Conner & Perlis (1984) p.16-17 We define addition and multiplication as follows: :(d_1,e_1,f_1) + (d_2,e_2,f_2) = ((-1)^d_1d_2, e_1+e_2, _1,d_2-d_1d_2,(-1)^]f_1f_2) :(d_1,e_1,f_1) \cdot (d_2,e_2,f_2) = (d_1^d_2^, e_1e_2, _1,d_2f_1^f_2^) \ . Then there is a surjective ring homomorphism from ''W''(''K'') to Sym(''K'') obtained by mapping a class to discriminant, rank mod 2, and the sequence of Hasse invariants. The kernel is ''I''3.Conner & Perlis (1984) p.18 The symbol ring is a realisation of the Brauer-Wall group.Lam (2005) p.116


Witt ring of the rationals

The Hasse–Minkowski theorem implies that there is an injectionLam (2005) p.174 : W(\mathbf) \rightarrow W(\mathbf) \oplus \prod_p W(\mathbf_p) \ . We make this concrete, and compute the image, by using the "second residue homomorphism" W(Q''p'') → W(F''p''). Composed with the map W(Q) → W(Q''p'') we obtain a group homomorphism ∂''p'': W(Q) → W(F''p'') (for ''p'' = 2 we define ∂2 to be the 2-adic valuation of the discriminant, taken mod 2). We then have a
split exact sequence In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way. Equivalent characterizations A short exact sequence of abelian groups or of modules over a ...
Lam (2005) p.175 : 0 \rightarrow \mathbf \rightarrow W(\mathbf) \rightarrow \mathbf/2 \oplus \bigoplus_ W(\mathbf_p) \rightarrow 0 \ which can be written as an isomorphism :W(\mathbf) \cong \mathbf \oplus \mathbf/2 \oplus \bigoplus_ W(\mathbf_p) \ where the first component is the signature.Lam (2005) p.178


Witt ring and Milnor's K-theory

Let ''k'' be a field of characteristic not equal to 2. The powers of the ideal ''I'' of forms of even dimension ("fundamental ideal") in W(k) form a descending
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter m ...
and one may consider the associated
graded ring 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 R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
, that is the direct sum of quotients I^n/I^. Let \langle a\rangle be the quadratic form ax^2 considered as an element of the Witt ring. Then \langle a\rangle - \langle 1\rangle is an element of ''I'' and correspondingly a product of the form : \langle\langle a_1,\ldots ,a_n\rangle\rangle = (\langle a_1\rangle - \langle 1\rangle)\cdots (\langle a_n\rangle - \langle 1\rangle) is an element of I^n.
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Univ ...
in a 1970 paper proved that the mapping from (k^*)^n to I^n/I^ that sends (a_1,\ldots ,a_n) to \langle\langle a_1,\ldots ,a_n\rangle\rangle is multilinear and maps Steinberg elements (elements such that for some i and j such that i\ne j one has a_i+a_j=1) to zero. This means that this mapping defines a homomorphism from the Milnor ring of ''k'' to the graded Witt ring. Milnor showed also that this homomorphism sends elements divisible by 2 to zero and that it is surjective. In the same paper he made a conjecture that this homomorphism is an isomorphism for all fields ''k'' (of characteristic different from 2). This became known as the Milnor conjecture on quadratic forms. The conjecture was proved by Dmitry Orlov, Alexander Vishik and
Vladimir Voevodsky Vladimir Alexandrovich Voevodsky (, russian: Влади́мир Алекса́ндрович Воево́дский; 4 June 1966 – 30 September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic va ...
in 1996 (published in 2007) for the case \textrm(k)=0, leading to increased understanding of the structure of quadratic forms over arbitrary fields.


Grothendieck-Witt ring

The Grothendieck-Witt ring ''GW'' is a related construction generated by isometry classes of nonsingular quadratic spaces with addition given by orthogonal sum and multiplication given by tensor product. Since two spaces that differ by a hyperbolic plane are not identified in ''GW'', the inverse for the addition needs to be introduced formally through the construction that was discovered by Grothendieck (see
Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic ...
). There is a natural homomorphism ''GW'' → Z given by dimension: a field is quadratically closed if and only if this is an isomorphism.Lam (2005) p. 34 The hyperbolic spaces generate an ideal in ''GW'' and the Witt ring ''W'' is the quotient.Lam (2005) p. 28 The
exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
gives the Grothendieck-Witt ring the additional structure of a
λ-ring In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λ''n'' on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide ...
.Garibaldi, Merkurjev & Serre (2003) p.63


Examples

* The Grothendieck-Witt ring of C, and indeed any
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
or
quadratically closed field In mathematics, a quadratically closed field is a field in which every element has a square root.Lam (2005) p. 33Rajwade (1993) p. 230 Examples * The field of complex numbers is quadratically closed; more generally, any algebraically clos ...
, is Z. * The Grothendieck-Witt ring of R is isomorphic to the group ring Z 'C''2 where ''C''2 is a cyclic group of order 2. * The Grothendieck-Witt ring of any finite field of odd characteristic is Z ⊕ Z/2Z with trivial multiplication in the second component.Lam (2005) p.36, Theorem 3.5 The element (1, 0) corresponds to the quadratic form ⟨''a''⟩ where ''a'' is not a square in the finite field. * The Grothendieck-Witt ring of a local field with maximal ideal of norm congruent to 1 modulo 4 is isomorphic to Z ⊕ (Z/2Z)3. * The Grothendieck-Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 it is Z ⊕ ''Z/4Z ⊕ Z/2Z.


Grothendieck-Witt ring and motivic stable homotopy groups of spheres

Fabien Morel Fabien Morel (born 22 January 1965, in Reims) is a French algebraic geometer and key developer of A¹ homotopy theory with Vladimir Voevodsky. Among his accomplishments is the proof of the Friedlander conjecture, and the proof of the complex case ...
showed that the Grothendieck-Witt ring of a
perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k'' i ...
is isomorphic to the motivic stable homotopy group of spheres π0,0(S0,0) (see "
A¹ homotopy theory In algebraic geometry and algebraic topology, branches of mathematics, homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schem ...
").


Witt equivalence

Two fields are said to be Witt equivalent if their Witt rings are isomorphic. For global fields there is a local-to-global principle: two global fields are Witt equivalent if and only if there is a bijection between their places such that the corresponding local fields are Witt equivalent. In particular, two number fields ''K'' and ''L'' are Witt equivalent if and only if there is a bijection ''T'' between the places of ''K'' and the places of ''L'' and a group isomorphism ''t'' between their square-class groups, preserving degree 2 Hilbert symbols. In this case the pair (''T'', ''t'') is called a reciprocity equivalence or a degree 2 Hilbert symbol equivalence. Some variations and extensions of this condition, such as "tame degree ''l'' Hilbert symbol equivalence", have also been studied.


Generalizations

Witt groups can also be defined in the same way for skew-symmetric forms, and for
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
s, and more generally
ε-quadratic form In mathematics, specifically the theory of quadratic forms, an ''ε''-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; , accordingly for symmetric or skew-symmetric. They are also called (-)^n-quadr ...
s, over any *-ring ''R''. The resulting groups (and generalizations thereof) are known as the even-dimensional symmetric ''L''-groups ''L''2''k''(''R'') and even-dimensional quadratic ''L''-groups ''L''2''k''(''R''). The quadratic ''L''-groups are 4-periodic, with ''L''0(''R'') being the Witt group of (1)-quadratic forms (symmetric), and ''L''2(''R'') being the Witt group of (−1)-quadratic forms (skew-symmetric); symmetric ''L''-groups are not 4-periodic for all rings, hence they provide a less exact generalization. ''L''-groups are central objects in
surgery theory In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while An ...
, forming one of the three terms of the
surgery exact sequence In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension >4. The surgery structure set \mathcal (X) of a compact n-dimensional manifold X i ...
.


See also

* Reduced height of a field


Notes


References

* * * * * * *


Further reading

* {{cite book , last=Balmer , first=Paul , chapter=Witt groups , editor1-last=Friedlander , editor1-first=Eric M. , editor2-last=Grayson , editor2-first=D. R. , title=Handbook of ''K''-theory , volume=2 , pages=539–579 , publisher=
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 ...
, year=2005 , isbn=3-540-23019-X , zbl=1115.19004


External links


Witt rings
in the Springer encyclopedia of mathematics Quadratic forms