Quadratic Forms and
Ordered Fields
Most readers will have met inner products before. Here we take up the subject in a
more general form and look at the properties of quadratic forms over a general field
and its group of isometries (Sections 8.1-8.3). With each quadratic form a certain
algebra is associated, the Clifford algebra, and with the set of all forms on a field
the Witt group is associated; these form the subject of Section 8.4 and Section 8.5
respectively, with a further development, the Witt ring of a field, in Section 8.9.
In Section 8.10 we take a brief look at symplectic groups and in Section 8.11 we
consider quadratic forms in characteristic 2. We also briefly discuss the related
topic of ordered fields in Section 8.6, leading to a construction of the real numbers
in Section 8.7 and formally real fields in Section 8.8. Some of the later topics are
included for completeness, but do not really have a place in a basic account; thus
at a first reading the later parts of Sections 8.7-8.11 can be omitted.
8.1 Inner Product Spaces
Let k be a field. By an inner product space we understand a pair (V, b) consisting of a
finite-dimensional vector space V over k and a symmetric bilinear form b on V, i.e.
a mapping
b: V x V -+ k,
which is symmetric: b(x, y) = bey, x), and bilinear:
b(ax + {3y, z) = ab(x, z) + {3b(y, z), b(x, ay + {3z) = ab(x, y) + {3b(x, z).
There is a second way of defining inner product spaces, in terms of quadratic forms.
By a quadratic form on a given vector space V over k we understand a mapping
q : V -+ k such that
Q.1 q(ax) = a 2 q(x) for all x E V and a E k,
Q.2 q(x + y) - q(x) - q(y) = bq(x, y) is a bilinear form on V.
The pair (V, q) is also called a quadratic space. By the dimension of q is meant the
dimension of the underlying space; we speak of a binary, ternary, or quaternary
form if the dimension is 2, 3 or 4.
249
,250 Quadratic Forms and Ordered Fields
Clearly the function bq in Q.2 is symmetric; it is called the associated bilinear form,
and it can be used to define the inner product structure on V. If char k =j:. 2, then
every symmetric bilinear form b is associated with a quadratic form, namely
b = bq, where q = !b(x, x). This follows because any bilinear form b on V satisfies
b(x + y, x + y) - b(x, x) - bey, y) = b(x, y) + bey, x).
When b is symmetric, this reduces to 2b(x, y), so in that case bq(x, y) =
q(x + y) - q(x) - q(y) = b(x, y). So in characteristic =j:. 2 there is complete equiva-
lence between quadratic forms and bilinear forms, and we shall use both inter-
changeably. To be precise, we shall use b(x, y) and q(x) = b(x, x), so that
b(x,y) =! [q(x + y) - q(x) - q(y)]. (8.1.1)
In characteristic 2 we can still associate with each quadratic form a bilinear form (as
in Q.2) and each quadratic form can be expressed as b(x, x) for some bilinear form b
(see Exercise 3), but now b cannot in general be chosen to be symmetric. In what
follows we shall concentrate on the case char k =j:. 2 and briefly look at the case
char k = 2 in Section 8.11.
Let V be an inner product space. Relative to the basis el, ... , en of V, its form b is
determined by the coefficients aij = b(ei, ej), for if x = L ~ie' y = L 17iei, then by
linearity we have b(x, y) = L aij~17. In matrix form this can be written as
b(x,y) = ~A17T, (8.1.2)
where ~ = (~I' ... , ~n)' 17 = (171, ... , 17n) and T denotes transposition. Since b is
symmetric, A is a symmetric matrix, i.e. AT = A. Conversely, any symmetric
matrix A defines a symmetric bilinear form on V by the formula (8.1.2).
If the basis of V is changed, let the new coordinates of x, y be the rows ~', 17', where
~ = ~'P, 17 = 17'P for some invertible matrix P (describing the change of basis in V).
Then b(x, y) = ~'A17T = ~'PApT 17'T. Since this holds for all ~', 17', we conclude that
A' = PApT, where P is invertible. (8.1.3)
Two matrices A, A' related as in (8.1.3) are said to be congruent; what we have said
shows that matrices of the form b in different coordinate systems are congruent.
Conversely, if b is represented by the matrix A in one coordinate system and A' is
congruent to A, say (8.1.3) holds, then transformation by P will define a new
basis, relative to which b has the matrix A'. This proves
Proposition 8.1.1. Two matrices represent the same bilinear form in different coordi-
nate systems if and only if they are congruent. •
The result holds for any bilinear forms, symmetric or not; for symmetric forms in
characteristic not 2, the matrices can also be chosen to be symmetric. Moreover,
given two matrices A, A', related as in (8.1.3), if one of them is symmetric, then
clearly so is the other.
,8.1 Inner Product Spaces 251
An isometry between two inner product spaces V, V' (or also between their forms)
is an isomorphism f : V -+ V I which transforms the form of V into that of V '; thus
if the forms are b, b' respectively, then
b'(xf,yf) = b(x,y) for all x,y E V.
Equivalently (in characteristic not 2), if the quadratic forms in V, V' are q, q' respec-
tively, then q'(xf) = q(x) for all x E V. We shall then say that the spaces V, V' are
isometric, and write V ~ V'. Relative to suitable bases the bilinear forms in spaces
that are isometric are represented by the same matrix; hence two spaces with
given bases are isometric iff the matrices of the forms relative to these bases are
congruent.
Let (V, b) be an inner product space. Any subspace U of V is again an inner
product space, the form being bl U, the restriction of the form b to U.
Two vectors x, y E V are said to be orthogonal if b(x, y) = 0; by the symmetry of b
this is a symmetric relation. For any subset S of V we define its orthogonal space as
S.L = {x E Vlb(x,y) = 0 for ally E S} . (8.1.4)
It is easily seen that S.L is a subspace of V; this holds for any subset S of V, although
we shall mainly use (8.1.4) when S is itself a subspace. In particular, the subspace V.L
is called the radical of V; it consists of all vectors orthogonal to all of V. If V.L ::f. 0,
then the form b (or also the space V) is called singular, otherwise V is non-singular or
regular. An inner product space V over the real numbers is called positive-definite if
its quadratic form q satisfies q(x) > 0 for all x ::f. 0; such a space is just the familiar
Euclidean space.
Let V be an inner product space whose form relative to a basis is given by (8.1.2).
The radical of V is obtained by solving the equations b(x, ei) = 0, i.e. xA = O. These
equations have a non-trivial solution precisely when A is singular, so we obtain
Proposition 8.1.2. An inner product space is singular if and only if the matrix of its
form (relative to any basis of the space) is singular. •
Let V be a regular space and U be any subspace. If VI, .. . , Vn is a basis of V, chosen
so that VI, ... , Vr is a basis of U, then a vector x = L aiVi is orthogonal to U iff
(8.1.5)
By hypothesis the n x n matrix (b(Vi, Vj)) is regular, hence the n x r matrix consist-
ing of the first r columns has rank r. Therefore the system (8.1.5) has rank r and the
space of solutions has dimension n - r. This proves the first part of
Proposition 8.1.3. If V is a regular inner product space and U is any subspace of V,
then
dim U + dim U.L = dim V (8.1.6)
and
U.L.L = U. (8.1.7)
, 252 Quadratic Forms and Ordered Fields
Proof. It only remains to prove (8.1.7). By (8.1.6) we have dim U + dim U1- =
dim U1- + dim U1-1-, hence dim U H = dim U. Since clearly Us:; U1-1-, it follows
that U1-1- = U. •
We note that un U1- need not be 0; whether it is will depend on whether U is
regular in the induced metric. We shall deal with this case in the next section.
For a regular space the determinant of the matrix, though not itself invariant,
provides an invariant of the space. By (8.1.3) we have the relation
det A' = det A. (det p)2. (8.1.8)
Here (det p)2 can assume the value of any non-zero square in the field k. If we denote
this set of squares by p2, then by (8.1.8) each regular inner product space V deter-
mines a unique element of the factor group P /P2, namely the residue class of det A.
This is called the determinant of the space V, or also of the form b.
Finally we remark that by fixing one of the arguments in an inner product, we
obtain a linear functional on the space, i.e. an element of V* = Homk(V, k). The
inner product b of V thus defines a mapping
CfJb : V -* V*, given by x 1-* b(x, -).
This mapping is linear, as is easily seen. It is injective iff b is regular; since V* and V
have the same dimension, CfJb is then an isomorphism.
Exercises
1. Let V be an inner product space. Show that for each u E V the mapping
Au : x 1-* b(x, u) is an element of V* and that the mapping CfJ: u 1-* Au is an
isomorphism iff V is regular. Examine the case where the coefficient ring is not
a field.
2. Let V be an inner product space and U I , U2 be subspaces. Show that (i)
UI s:; U2 ::::} ut ;2 uf-, (ii) U I s:; ut1-, (iii) ut = ut 1-1-.
3. Given a quadratic form q on a space V in characteristic 2, find a bilinear form h on
V such that q(x) = hex, x). Find the conditions on q for which h can be chosen
symmetric as well as bilinear. (Hint. See Section 8.11.)
4. Show that any quadratic form q satisfies the parallelogram law: q(x + y) +
q(x - y) = 2[q(x) + q(y)].
5. Let V be a regular inner product space and U I , U2 be subspaces. Show that
(U I + U2 )1- = ut n Uf-, (U I n U2 )1- = ut + U{
8.2 Orthogonal Sums and Diagonalization
Let V, V' be inner product spaces with quadratic forms q, q' respectively; their ortho-
gonal sum is defined as the direct sum of the spaces Vand V I with the quadratic form
Q defined by
Q(x + x') = q(x) + ql(X'), X E V, x' E Vi (8.2.1)
Ordered Fields
Most readers will have met inner products before. Here we take up the subject in a
more general form and look at the properties of quadratic forms over a general field
and its group of isometries (Sections 8.1-8.3). With each quadratic form a certain
algebra is associated, the Clifford algebra, and with the set of all forms on a field
the Witt group is associated; these form the subject of Section 8.4 and Section 8.5
respectively, with a further development, the Witt ring of a field, in Section 8.9.
In Section 8.10 we take a brief look at symplectic groups and in Section 8.11 we
consider quadratic forms in characteristic 2. We also briefly discuss the related
topic of ordered fields in Section 8.6, leading to a construction of the real numbers
in Section 8.7 and formally real fields in Section 8.8. Some of the later topics are
included for completeness, but do not really have a place in a basic account; thus
at a first reading the later parts of Sections 8.7-8.11 can be omitted.
8.1 Inner Product Spaces
Let k be a field. By an inner product space we understand a pair (V, b) consisting of a
finite-dimensional vector space V over k and a symmetric bilinear form b on V, i.e.
a mapping
b: V x V -+ k,
which is symmetric: b(x, y) = bey, x), and bilinear:
b(ax + {3y, z) = ab(x, z) + {3b(y, z), b(x, ay + {3z) = ab(x, y) + {3b(x, z).
There is a second way of defining inner product spaces, in terms of quadratic forms.
By a quadratic form on a given vector space V over k we understand a mapping
q : V -+ k such that
Q.1 q(ax) = a 2 q(x) for all x E V and a E k,
Q.2 q(x + y) - q(x) - q(y) = bq(x, y) is a bilinear form on V.
The pair (V, q) is also called a quadratic space. By the dimension of q is meant the
dimension of the underlying space; we speak of a binary, ternary, or quaternary
form if the dimension is 2, 3 or 4.
249
,250 Quadratic Forms and Ordered Fields
Clearly the function bq in Q.2 is symmetric; it is called the associated bilinear form,
and it can be used to define the inner product structure on V. If char k =j:. 2, then
every symmetric bilinear form b is associated with a quadratic form, namely
b = bq, where q = !b(x, x). This follows because any bilinear form b on V satisfies
b(x + y, x + y) - b(x, x) - bey, y) = b(x, y) + bey, x).
When b is symmetric, this reduces to 2b(x, y), so in that case bq(x, y) =
q(x + y) - q(x) - q(y) = b(x, y). So in characteristic =j:. 2 there is complete equiva-
lence between quadratic forms and bilinear forms, and we shall use both inter-
changeably. To be precise, we shall use b(x, y) and q(x) = b(x, x), so that
b(x,y) =! [q(x + y) - q(x) - q(y)]. (8.1.1)
In characteristic 2 we can still associate with each quadratic form a bilinear form (as
in Q.2) and each quadratic form can be expressed as b(x, x) for some bilinear form b
(see Exercise 3), but now b cannot in general be chosen to be symmetric. In what
follows we shall concentrate on the case char k =j:. 2 and briefly look at the case
char k = 2 in Section 8.11.
Let V be an inner product space. Relative to the basis el, ... , en of V, its form b is
determined by the coefficients aij = b(ei, ej), for if x = L ~ie' y = L 17iei, then by
linearity we have b(x, y) = L aij~17. In matrix form this can be written as
b(x,y) = ~A17T, (8.1.2)
where ~ = (~I' ... , ~n)' 17 = (171, ... , 17n) and T denotes transposition. Since b is
symmetric, A is a symmetric matrix, i.e. AT = A. Conversely, any symmetric
matrix A defines a symmetric bilinear form on V by the formula (8.1.2).
If the basis of V is changed, let the new coordinates of x, y be the rows ~', 17', where
~ = ~'P, 17 = 17'P for some invertible matrix P (describing the change of basis in V).
Then b(x, y) = ~'A17T = ~'PApT 17'T. Since this holds for all ~', 17', we conclude that
A' = PApT, where P is invertible. (8.1.3)
Two matrices A, A' related as in (8.1.3) are said to be congruent; what we have said
shows that matrices of the form b in different coordinate systems are congruent.
Conversely, if b is represented by the matrix A in one coordinate system and A' is
congruent to A, say (8.1.3) holds, then transformation by P will define a new
basis, relative to which b has the matrix A'. This proves
Proposition 8.1.1. Two matrices represent the same bilinear form in different coordi-
nate systems if and only if they are congruent. •
The result holds for any bilinear forms, symmetric or not; for symmetric forms in
characteristic not 2, the matrices can also be chosen to be symmetric. Moreover,
given two matrices A, A', related as in (8.1.3), if one of them is symmetric, then
clearly so is the other.
,8.1 Inner Product Spaces 251
An isometry between two inner product spaces V, V' (or also between their forms)
is an isomorphism f : V -+ V I which transforms the form of V into that of V '; thus
if the forms are b, b' respectively, then
b'(xf,yf) = b(x,y) for all x,y E V.
Equivalently (in characteristic not 2), if the quadratic forms in V, V' are q, q' respec-
tively, then q'(xf) = q(x) for all x E V. We shall then say that the spaces V, V' are
isometric, and write V ~ V'. Relative to suitable bases the bilinear forms in spaces
that are isometric are represented by the same matrix; hence two spaces with
given bases are isometric iff the matrices of the forms relative to these bases are
congruent.
Let (V, b) be an inner product space. Any subspace U of V is again an inner
product space, the form being bl U, the restriction of the form b to U.
Two vectors x, y E V are said to be orthogonal if b(x, y) = 0; by the symmetry of b
this is a symmetric relation. For any subset S of V we define its orthogonal space as
S.L = {x E Vlb(x,y) = 0 for ally E S} . (8.1.4)
It is easily seen that S.L is a subspace of V; this holds for any subset S of V, although
we shall mainly use (8.1.4) when S is itself a subspace. In particular, the subspace V.L
is called the radical of V; it consists of all vectors orthogonal to all of V. If V.L ::f. 0,
then the form b (or also the space V) is called singular, otherwise V is non-singular or
regular. An inner product space V over the real numbers is called positive-definite if
its quadratic form q satisfies q(x) > 0 for all x ::f. 0; such a space is just the familiar
Euclidean space.
Let V be an inner product space whose form relative to a basis is given by (8.1.2).
The radical of V is obtained by solving the equations b(x, ei) = 0, i.e. xA = O. These
equations have a non-trivial solution precisely when A is singular, so we obtain
Proposition 8.1.2. An inner product space is singular if and only if the matrix of its
form (relative to any basis of the space) is singular. •
Let V be a regular space and U be any subspace. If VI, .. . , Vn is a basis of V, chosen
so that VI, ... , Vr is a basis of U, then a vector x = L aiVi is orthogonal to U iff
(8.1.5)
By hypothesis the n x n matrix (b(Vi, Vj)) is regular, hence the n x r matrix consist-
ing of the first r columns has rank r. Therefore the system (8.1.5) has rank r and the
space of solutions has dimension n - r. This proves the first part of
Proposition 8.1.3. If V is a regular inner product space and U is any subspace of V,
then
dim U + dim U.L = dim V (8.1.6)
and
U.L.L = U. (8.1.7)
, 252 Quadratic Forms and Ordered Fields
Proof. It only remains to prove (8.1.7). By (8.1.6) we have dim U + dim U1- =
dim U1- + dim U1-1-, hence dim U H = dim U. Since clearly Us:; U1-1-, it follows
that U1-1- = U. •
We note that un U1- need not be 0; whether it is will depend on whether U is
regular in the induced metric. We shall deal with this case in the next section.
For a regular space the determinant of the matrix, though not itself invariant,
provides an invariant of the space. By (8.1.3) we have the relation
det A' = det A. (det p)2. (8.1.8)
Here (det p)2 can assume the value of any non-zero square in the field k. If we denote
this set of squares by p2, then by (8.1.8) each regular inner product space V deter-
mines a unique element of the factor group P /P2, namely the residue class of det A.
This is called the determinant of the space V, or also of the form b.
Finally we remark that by fixing one of the arguments in an inner product, we
obtain a linear functional on the space, i.e. an element of V* = Homk(V, k). The
inner product b of V thus defines a mapping
CfJb : V -* V*, given by x 1-* b(x, -).
This mapping is linear, as is easily seen. It is injective iff b is regular; since V* and V
have the same dimension, CfJb is then an isomorphism.
Exercises
1. Let V be an inner product space. Show that for each u E V the mapping
Au : x 1-* b(x, u) is an element of V* and that the mapping CfJ: u 1-* Au is an
isomorphism iff V is regular. Examine the case where the coefficient ring is not
a field.
2. Let V be an inner product space and U I , U2 be subspaces. Show that (i)
UI s:; U2 ::::} ut ;2 uf-, (ii) U I s:; ut1-, (iii) ut = ut 1-1-.
3. Given a quadratic form q on a space V in characteristic 2, find a bilinear form h on
V such that q(x) = hex, x). Find the conditions on q for which h can be chosen
symmetric as well as bilinear. (Hint. See Section 8.11.)
4. Show that any quadratic form q satisfies the parallelogram law: q(x + y) +
q(x - y) = 2[q(x) + q(y)].
5. Let V be a regular inner product space and U I , U2 be subspaces. Show that
(U I + U2 )1- = ut n Uf-, (U I n U2 )1- = ut + U{
8.2 Orthogonal Sums and Diagonalization
Let V, V' be inner product spaces with quadratic forms q, q' respectively; their ortho-
gonal sum is defined as the direct sum of the spaces Vand V I with the quadratic form
Q defined by
Q(x + x') = q(x) + ql(X'), X E V, x' E Vi (8.2.1)
