FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 3 definitions, 4 theorems, 2 propositions, 3 corollaries, 3 examples and 4 remarks.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Theorem 8.1.1
Theorem 8.1.1 (Diagonalization; \(\mathrm { c h a r } ( k ) \neq 2 ,\) ). Every non-degenerate symmetric bilinear form over a field \(k\) of characteristic different from 2 is isometric to a diagonal form \[ \langle a _ { 1 } \rangle \oplus \cdots \oplus \langle a _ { n } \rangle , w i t h a _ { j } \in k ^ { \times } , j = 1 , \ldots , n . \]
Remark 8.1.2
Remark 8.1.2. 1. For the field \(\mathbb { R }\) we may further simplify this: since every positive number is a square, we may assume that \(a _ { j } \in \{ 1 , - 1 \}\) . The number of times 1 or \(^ { - 1 }\) appears is unique as stated by Sylvester’s theorem. Its proof is recalled below (see Corollary 8.1.3). 2. For an algebraically closed field such as the field \(\mathbb { C }\) we may assume that the \(a _ { j }\) are all equal to 1, i.e., every form is equivalent to a sum of squares. What about uniqueness of the above representations? The Witt decomposition \(V = U ^ { \circledast m } \circledast V ^ { \prime }\) , where \(V ^ { \prime }\) is totally anisotropic, has the merit of being unique. Over \(\mathbb { R }\) and \(\mathbb { C }\) the diagonal decomposition of \(V ^ { \prime }\) is also unique. It is instructive to compare this with what we said about the Witt ring of \(\mathbb { C }\) and \(\mathbb { R }\) in Example 7.3.2. In Section 8.3 we show uniqueness over finite fields. However, in general we have no uniqueness, as explained in Chapter 3 where we investigated this for the field \(\mathbb { Q }\) .
Corollary 8.1.3
Corollary 8.1.3 (Sylvester’s Law). Let \(( V , b )\) be a real inner product space. There is a subspace \(V ^ { + }\) (respectively \(V ^ { - }\) ) of maximal dimension \(d ^ { + }\) (respectively \(d ^ { - }\) ) on which \(^ { b }\) is positive (negative) such that \[ V = V ^ { + } \oplus V ^ { - } . \] Moreover, all such spaces \(V ^ { \pm }\) have the same dimension \(d ^ { \pm }\) .
Definition 8.1.4
Definition 8.1.4. Referring to (8.1), the pair \(( \dim V ^ { + } , \dim V ^ { - } )\) is called the signature of the inner product space \(V\) over \(\mathbb { R }\) . The difference \(\tau ( V ) = \dim V ^ { + } - \dim V ^ { - }\) is called the index . If \(V = V ^ { + }\) , the inner product space is called positive definite, if \(V = V ^ { - }\) we say that \(V\) is negative definite.
Theorem 8.1.5
Theorem 8.1.5 (Classification of real forms). Two real inner product spaces \(( V , b )\) and \(( V ^ { \prime } , b ^ { \prime } )\) are isometric if and only if \(\dim V = \dim V ^ { \prime }\) and \(\tau ( V ) = \tau ( V ^ { \prime } )\) .
Proposition 8.2.1
Proposition 8.2.1. Let \(( V , b )\) be a symmetric inner product space over a field \(k\) of characteristic 2. Then \(V\) is isometric to an orthogonal direct sum \(V _ { 1 } \odot V _ { 2 }\) , where \(V _ { 1 }\) is a diagonal form and \(V _ { 2 }\) is a direct sum of hyperbolic planes.
Definition / construction
The Arf invariant of a non-degenerate quadratic form in characteristic \(2\) is the class obtained from a symplectic basis by the standard sum \(\sum q(e_i)q(f_i)\) modulo the Artin–Schreier subgroup.
Definition / construction / notation
no longer depends on the particular symplectic basis. It is called the Arf invariant of \(q\) :
Theorem 8.2.2
Theorem 8.2.2. A non-degenerate quadratic form on \(V\) in characteristic 2 is equivalent to a form \(\begin{array} { r } { q = \sum _ { i = 1 } ^ { n } x _ { i } x _ { n + i } + \sum _ { j = 1 } ^ { 2 n } a _ { j } x _ { j } ^ { 2 } } \end{array}\) , for some \(a _ { j } \in k\) . 1. The Arf invariant of such a form is well defined. 2. Isometric non-degenerate quadratic forms have the same Arf invariant. 3. If \(k\) is a perfect field of characteristic 2, then two non-degenerate quadratic forms of the same rank and the same Arf invariant are isometric.
Corollary 8.2.3
Corollary 8.2.3. Let \(( V , q )\) be a quadratic inner product space of rank \(2 n\) over a perfect field of characteristic 2. Then coordinates may be chosen so that \(q\) is equivalent to a form with Arf invariant \(c ^ { 2 } \equiv c\) mod \(\wp ( k )\) given by \[ q ^ { ( c ) } ( x ) : = \sum _ { i = 1 } ^ { n } x _ { i } x _ { n + i } + c ( x _ { n } ^ { 2 } + x _ { 2 n } ^ { 2 } ) \ = \sum _ { i = 1 } ^ { n - 1 } x _ { i } x _ { n + i } + x _ { n } x _ { 2 n } + c ( x _ { n } ^ { 2 } + x _ { 2 n } ^ { 2 } ) , \quad c \in k . \] This form is isometric to \(\mathbb { Q } ^ { n - 1 } U \mathbb { Q } P ^ { ( c ) }\) where the binary form on \(P ^ { ( c ) }\) is given by \(x y + c ( x ^ { 2 } + y ^ { 2 } )\) .
Example / comparison
Example 8.2.4. The cyclic group \(\mathbb { Z } / 2 \mathbb { Z }\) can also be viewed as the field \(\mathbb { F } _ { 2 }\) and the \({ \frac { 1 } { 2 } } \mathbb { Z } / \mathbb { Z }\) -valued torsion forms \(u _ { 1 }\) and \(v _ { 1 }\) from Examples 1.9.5 give \(\mathbb { F } _ { 2 }\) -valued forms \(U\) and \(V\) with Arf invariants \(0\) , \(^ { 1 }\) , respectively. So for \(k = \mathbb { F } _ { 2 }\) the only two types of rank \(2 n\) non-degenerate quadratic forms are \(U ^ { \left( \perp \right) n }\) and \(U ^ { \oplus ( n - 1 ) } \oplus V\) . Below, in Section 8.3, we generalize this for all finite fields.
Proposition 8.2.5
Proposition 8.2.5. Let \(k\) be a field of characteristic 2. 1. A \(k\) -vector space \(V\) with a symmetric bilinear form \(b\) is isometric to \(\mathbb { O } _ { i = 1 } ^ { s } \langle a _ { i } \rangle \mathbb { O }\) \(U ^ { \oplus t } \oplus \operatorname { r a d } ( b )\) , \(a _ { i } \in k ^ { \times }\) , \(i = 1 , \dots , s\) . 2. A \(k\) -vector space \(V\) with a quadratic form \(q\) is isometric to \(q ^ { \prime } \textcircled { \div } \mathrm { r a d } ( b _ { q } )\) where (a) \(q ^ { \prime }\) n-degenera, for some dratic form equivalent to a form . \(\textstyle \sum _ { i = 1 } ^ { m } x _ { i } x _ { m + i } +\) \(\textstyle \sum _ { j = 1 } ^ { 2 m } a _ { j } x _ { j } ^ { 2 }\) \(a _ { j } \in k\) (b) \() \mathrm { ~ r a d } ( b _ { q } ) = V _ { q } \textcircled { \div } \mathrm { r a d } ( q )\) , with \(q | _ { V _ { q } }\) diagonal of rank equal to \(\dim V _ { q } = s\) , \(s \leq [ k : k ^ { 2 } ]\) . If \(k\) is a perfect field (e.g. if \(k = \mathbb { F } _ { 2 ^ { r } }\) , a finite field, or \(k\) is algebraically closed), then (a) \(q ^ { \prime } \simeq U ^ { \oplus ( m - 1 ) } \oplus P ^ { ( c ) }\) . (b) \(s = 0\) or \(s = 1\) . Hence \(\operatorname { r a d } ( b _ { q } ) = \operatorname { r a d } ( q )\) or \(\operatorname { r a d } ( b _ { q } ) = \langle a \rangle \oplus \operatorname { r a d } ( q )\) , \(a \in k ^ { \times }\) . In the last case, \(i f \operatorname { r a d } ( q ) = 0\) , \(q\) is semi-unimodular.
Remark 8.2.6
Remark 8.2.6. Note that if \(\operatorname { r a d } ( q ) = 0\) and \(\dim V = s\) is odd, the semi-discriminant sdisc(??) (see Definition 6.3.3) is non-zero precisely if \(s = 1\) . Hence if \(k\) is perfect and \(b _ { q } = 0\) , but \(q\) is non-degenerate, necessarily \(s = 1\) and the form is semi-unimodular.
Example / comparison
Example 8.2.7. Over \(\mathbf F_2\), non-degenerate even-dimensional quadratic spaces have the two isometry types distinguished by Arf invariant \(0\) or \(1\); in odd dimension, a semi-unimodular form is the corresponding even-dimensional normal form plus one square term.
Example / comparison
Example 8.3.1. Over a finite field \(k\), every totally anisotropic binary quadratic form is isometric to a scalar norm form for the unique quadratic extension \(K/k\). In characteristic \(2\), writing \(K=k(\xi)\) with \(\xi^2+\xi+c=0\), irreducibility is equivalent to \(\operatorname{Tr}_{k/\mathbf F_2}(c)=1\), and the norm form is \(x^2+xy+cy^2\).
Remark 8.3.2
Remark 8.3.2. We just explained that to classify quadratic spaces over a finite field of characteristic 2, one needs to specify an element with absolute trace equal to 1 and that for \(k = \mathbb { F } _ { 2 ^ { r } }\) with \(r\) odd, one can take \(c = 1\) . To see what happens for \(r\) even, let us first consider the case of \(k\) iterated quadratic extensions, say \(k _ { 0 } = \mathbb { F } _ { 2 } \subset k _ { 1 } = \mathbb { F } _ { 2 } ( s _ { 1 } ) \subset \cdots \subset k _ { m } = k _ { m - 1 } ( s _ { m } )\) , where \(s _ { j }\) is a root of an Artin– Schreier polynomial \(X ^ { 2 } + X + c _ { j }\) . We claim that the product \(c _ { 1 } \cdots c _ { m }\) has absolute trace 1. To show this, note that \(\mathsf { T r } _ { k _ { j } / k _ { j - 1 } } ( c _ { j } ) = 1\) by the choice of \(c _ { j }\) so that recursively \[ \begin{array} { r l } & { \mathsf { T r } _ { k _ { m } / k _ { 0 } } ( c _ { 1 } \cdot \cdot \cdot c _ { m } ) = \mathsf { T r } _ { k _ { m } / k _ { m - 1 } } ( c _ { m } ) \cdot \mathsf { T r } _ { k _ { m - 1 } / k _ { 0 } } ( c _ { 1 } \cdot \cdot \cdot c _ { m - 1 } ) } \\ & { \phantom { \frac { 1 } { 1 } } = \mathsf { T r } _ { k _ { m - 1 } / k _ { 0 } } ( c _ { 1 } \cdot \cdot \cdot c _ { m - 1 } ) = 1 . } \end{array} \] For the general case \(r = 2 ^ { m } s\) with \(s\) odd, we simply write \(k\) as an extension of \(\mathbb { F } _ { 2 ^ { m } }\) . Since \(\deg ( k / \mathbb { F } _ { 2 ^ { m } } ) = s\) is odd, \(\mathsf { T r } _ { k / \mathbb { F } _ { 2 } } ( c ) = s \mathsf { T r } _ { \mathbb { F } _ { 2 ^ { m } } / \mathbb { F } _ { 2 } } ( c ) = 1\) as well.
Theorem 8.3.3
Theorem 8.3.3 (Classification of quadratic inner product spaces over finite fields). Let \(k\) be a finite field of characteristic \(p\) , \(K\) the unique quadratic extension field of \(k\) and let \(V\) be a quadratic inner product space over \(k\) . \(p \neq 2\) • \(\dim V\) even. Then \(V \simeq U ^ { \oplus m }\) or \(V \simeq U ^ { \oplus m } \oplus \mathsf { N } _ { K / k }\) . The form \(\mathsf { N } _ { K / k }\) is isometric to a diagonal form \(\left. 1 \right. \ @ \left. - \varepsilon \right.\) with ?? a non-square. • \(\dim V\) odd. Then \(V \simeq U ^ { \oplus m } \oplus \langle c \rangle\) and there are two isometry classes according to whether \(c \neq 0\) is a square or not. In both cases, the given decomposition is the Witt decomposition and so the Witt index takes only the values \({ \frac { 1 } { 2 } } \dim V\) , \({ \frac { 1 } { 2 } } \left( \dim V - 1 \right)\) or \({ \scriptstyle { \frac { 1 } { 2 } } } ( \dim V - 2 )\) . \(p = 2\) Then dim ?? is even, and either \(V \simeq U ^ { \oplus m }\) with zero Arf invariant, or \(V \simeq\) \(U ^ { \oplus m } \oplus \mathsf { N } _ { K / k }\) with non-zero Arf invariant. More precisely, if \(K = k ( \xi )\) with \(\xi ^ { 2 } + \xi + c = 0\) , then \(\mathsf { N } _ { K / k }\) is isometric to the binary form \(c x ^ { 2 } + x y + c y ^ { 2 }\) with Arf invariant \(c ^ { 2 } \equiv c\) mod \(\wp ( k )\) .
Corollary 8.3.4
Corollary 8.3.4. If \(p \neq 2\) every quadratic inner product space \(( V , q )\) over \(\mathbb { F } _ { p ^ { r } }\) of rank ?? is isometric to \(\langle 1 \rangle ^ { \circled { \ 4 } m }\) if \(\operatorname { d i s c } ( q ) = 1\) and to \(\langle 1 \rangle ^ { \oplus m - 1 } \oplus \langle \epsilon \rangle\) if \(\operatorname { d i s c } ( q ) = \epsilon\) , \(a\) non-square modulo \(\mathbb { F } _ { p ^ { r } }\) .
Remark 8.3.5
Remark 8.3.5. Recall that in characteristic 2 several non-isometric quadratic forms have isometric polar forms and so the classification of symmetric inner product spaces is simpler than that of quadratic inner product spaces: Proposition 8.2.1 implies that inner product spaces over \(\Psi _ { 2 ^ { r } }\) are isometric to \(\langle u ^ { ( 1 ) } \rangle \mathbb { \oplus \cdot \cdot \cdot } \mathbb { \oplus } \langle u ^ { ( a ) } \rangle \mathbb { \oplus } U ^ { \oplus b }\) , where \(u ^ { ( j ) } \in \mathbb { F } _ { 2 ^ { r } } ^ { \times }\) , \(j = 1 , \dotsc , a\) .