Dependencies

Let \(K\) be a field and \(Q\) and \(Q'\) be nondegenerate quadratic forms over \(K\) of rank at least \(1\). Then the following properties are equivalent:

1. \(Q \dotminus Q'\) represents \(0\).

2. There exists \(r \in K^\times \) which is represented by \(Q\) and \(Q'\).

3. There exists \(r \in K^\times \) such that \(Q \dotminus rZ^2\) and \(Q' \dotminus rZ^2\) represent \(0\).

Let \(K\) be a field, let \(r \in K^\times \) and let \(Q\) be a quadratic form on a free \(K\)-vector space \(V\). Then \(Q\) represents \(r\) if and only if the form \(Q' := Q \dotminus rZ\) represents \(0\).

A sequence of bases \((b^{(0)} = b, \cdots , b^{(m)} = b')\) as in Theorem 3.1.7 is called a chain of orthogonal bases contiguously relating \(b\) to \(b'\).

Two bases of \(V\) are called contiguous if they have an element in common.

Let \(p\) be a prime number and let \(f\) be a quadratic form of rank \(n\) over \(\mathbb {Q}_p\). Take a diagonal quadratic form \(a_1X_1^2 + \cdots + a_nX_n^2\) which is equivalent to \(f\); then the discriminant of \(f\) is the element \(d(f) = a_1 \cdots a_n \in \mathbb {Q}_p^\times /{\mathbb {Q}_p^\times }^2\).

Note that Mathlibcontains QuadraticForm.discr, as an element of \(\mathbb {Q}_p\). We might need to adjust some statements and proofs if we really need to regard it as an element in \(\mathbb {Q}_p^\times /{\mathbb {Q}_p^\times }^2\).

Let \(u \in (\mathbb {Z}_2)^\times \). We define \(\epsilon (u)\) to be the class modulo \(2\) of \(\dfrac {u-1}{2}\).

A quadratic form \(f\) with coefficients in \(\mathbb {Q}\) is everywhere locally isotropic if \(f_v\) is isotropic for all \(v \in V\).

Let \(S\) be a finite set of prime numbers. We define the finite embedding of \(\mathbb {Q}\) into \(\mathbb {R}\times \prod _{p \in S} \mathbb {Q}_p\) as the diagonal map.

Let \(k\) be a field and \(a, b \in k\). If both \(a,b\) are nonzero, the Hilbert symbol \((a, b)_k\) is defined as follows: if there exist \(x, y, z \in k\), not all zero, such that \(z^2 = ax^2 + by^2\), then \((a, b)_k = 1\); otherwise, \((a, b)_k = -1\). If one of \(a,b\) is zero, we set \((a, b)_k = 0\).

Let \(p\) be a prime number and let \(f\) be a quadratic form of rank \(n\) over \(\mathbb {Q}_p\). Take a diagonal quadratic form \(a_1X_1^2 + \cdots + a_nX_n^2\) which is equivalent to \(f\); then the Hasse–Minkowski invariant of \(f\) is the element

A quadratic form is hyperbolic if it is equivalent to the form \(X^2 - Y^2\).

A quadratic form \(f : M \to A\) is called isotropic if there exists a nonzero \(x \in M\) such that \(f(x) = 0\).

Let \(p\) be a prime and \(u \in \mathbb {Z}_p^\times \). The Legendre symbol \(\left(\dfrac {u}{p}\right)\) is defined as

where \(\overline{u}\) is the image of \(u\) under the natural map \(\mathbb {Z}_p^\times \to (\mathbb {Z}/p\mathbb {Z})^\times \).

Let \(u \in (\mathbb {Z}_2)^\times \). We define \(\omega (u)\) to be the class modulo \(2\) of \(\dfrac {u^2-1}{8}\).

Let \(p\) be a prime and \(x \in \mathbb {Q}_p^\times \). The unit part of \(x\) is defined as the element of \(\mathbb {Z}_p^\times \) corresponding to \(x \cdot p^{-v_p(x)}\).

Let \(R\) be a ring. An element \((x_1, \cdots , x_n) \in R^n\) is primitive if at least one of the \(x_i\) is invertible.

Let \(M\) be a module over a commutative ring \(A\). A function \(f : M \to A\) is called a quadratic form on \(M\) if

1. \(f(ax) = a^2f(x)\) for all \(a \in A, x \in M\),

2. The function \((x, y) \mapsto f(x + y) - f(x) - f(y)\) is a bilinear form.

Let \(R\) be a commutative ring, let \(V\) be an \(R\)-module and let \(Q\) be a quadratic form on \(V\). We say that \(Q\) represents an element \(r \in R\) if there exists \(x \in V\) such that \(Q(v) = r\) and \(v \ne 0\).

Let \(a, b \in k^\times \). Then \((a, b)_k = 1\) if and only if there exists \(t \in k(\sqrt{b})\) such that \(a = N_{k(\sqrt{b})/k}(t)\).

Let \(K\) be a field of characteristic not equal to \(2\), let \(M\) be a \(K\)-vector space of positive dimension \(n\) and let \(f : M \to K\) be a nondegenerate quadratic form. Then there exists a form \(X_1^2 + a_2X_2^2 + \cdots + a_nX_n^2\) for some \(a_i \in K^\times \) such that it is isotropic if and only if \(f\) is.

Let \(K\) be a field of characteristic not equal to \(2\), let \(M\) be a \(K\)-vector space of positive dimension \(n\) and let \(f : M \to K\) be a nondegenerate quadratic form. Then there exists a form \(X_1^2 + a_2X_2^2 + \cdots + a_nX_n^2\), for some squarefree \(a_i \in K^\times \) such that it is isotropic if and only if \(f\) is.

Let \(f\) be an isotropic quadratic form over \(\mathbb {Q}\). Then \(f\) is everywhere locally isotropic.

Let \(V\) be a \(K\)-vector space of dimension at least \(3\), let \(Q\) be a nondegenerate quadratic form on \(V\), and let \(b, b'\) be two \(Q\)-orthogonal bases of \(V\). Assume that

There exists \(x \in K\) such that \(b_x := b_1' + x b_2'\) is nonisotropic and generates with \(b_1\) a nondegenerate plane.

The value \(\epsilon (f)\) does not depend on the choice of diagonal quadratic form equivalent to \(f\).

Let \(f\) be a degenerate quadratic form over \(\mathbb {Q}\). Then \(f\) is isotropic if and only if \(f\) is everywhere locally isotropic. In other words, \(f\) has a global zero if and only if \(f\) has everywhere a local zero.

Let \(f\) be a quadratic form over \(\mathbb {Q}\) of rank \(n \ge 5\) which is everywhere locally isotropic. Then \(f\) is isotropic.

Let \(f\) be a quadratic form over \(\mathbb {Q}\) of rank four which is everywhere locally isotropic. Then \(f\) is isotropic.

Let \(f\) be a quadratic form over \(\mathbb {Q}\) of rank one which is everywhere locally isotropic. Then \(f\) is isotropic.

Let \(f\) be a quadratic form over \(\mathbb {Q}\) of rank three which is everywhere locally isotropic. Then \(f\) is isotropic.

Let \(f\) be a quadratic form over \(\mathbb {Q}\) of rank two which is everywhere locally isotropic. Then \(f\) is isotropic.

Let \(f\) be a quadratic form over \(\mathbb {Q}\) of rank zero which is everywhere locally isotropic. Then \(f\) is isotropic.

Let \(p\) be a prime, let \(f \in \mathbb {Z}_p[X_1, \cdots , X_m]\) be a polynomial with \(p\)-adic integer coefficients, and let \(x \in (\mathbb {Z}/p\mathbb {Z})^m\) be a zero of \(f\) modulo \(p^n\), for some positive integer \(n\). Suppose that, for some \(j \in \{ 1, \cdots , m\} \), the partial derivative \(\frac{\partial f}{\partial X_j}(x)\) is nonzero modulo \(p^k\), where \(0 {\lt} 2k {\lt} n\). Then there exists a zero \(z\) of \(f\) in \(\mathbb {Z}_p^m\) whose reduction modulo \(p^n\) is \(x\), and such that \(z-x\) is divisible by \(p^{n-k}\).

Let \(p\) be a prime and let \(v \in (\mathbb {Q}_p)^\times \). Let \(x, y, z \in \mathbb {Q}_p\) be such that \((x, y, z) \ne (0, 0, 0)\) and \(z^2 - p x^2 - v y^2 = 0\). Then there exist \(z', y' \in (\mathbb {Z}_p)^\times \) and \(x' \in \mathbb {Z}_p\) such that \((z', y', x')\) is a nontrivial solution to the same equation.

Let \(p\) be a prime, let \(x \in \mathbb {Q}_p^\times \), and let \(v_p\) denote the \(p\)-adic valuation. Then \(\| x \cdot p^{-v_p(x)}\| = 1\).

Let \(Q\) and \(Q'\) be two equivalent quadratic forms over \(R\) and let \(r \in \). Then \(Q\) represents \(r\) if and only if \(Q'\) represents \(r\). In particular, \(Q\) is isotropic if and only if \(Q'\) is isotropic.

Let \(k\) be either \(\mathbb {Q}_p\) or \(\mathbb {R}\). Let \(f\) be a quadratic form of rank \(2\) over \(k\), and let \(a \in k^\times / {k^\times }^2\). Then \(f\) represents \(a\) if and only if \((a, -d(f)) = \epsilon (f)\).

Let \(R\) be a commutative ring, let \(V\) be an \(R\)-module and let \(Q\) be a quadratic form on \(V\). Then \(Q\) represents \(0\) if and only if it is isotropic.

Let \(k\) be either \(\mathbb {Q}_p\) or \(\mathbb {R}\). Let \(f\) be a quadratic form of rank \(3\) over \(k\). Then \(f\) represents \(0\) if and only if \((-1, -d(f)) = \epsilon (f)\).

Let \(a, b, a', b' \in k^\times \). Then \((a a'^2, b b'^2)_k = (a,b)_k\).

The quadratic form \(XY\) is hyperbolic.

Let \(f^{(i)} \in \mathbb {Z}_p[X_1, \cdots , X_m]\) be homogeneous polynomials with \(p\)-adic integer coefficients. For each \(n \ge 1\), denote by \(f_n^{(i)}\) the reduction of \(f^{(i)}\) modulo \(p^n\). The following are equivalent:

4. The \(f^{(i)}\) have a nontrivial common zero in \(\mathbb {Q}_p^m\).

5. The \(f^{(i)}\) have a common primitive zero in \(\mathbb {Z}_p^m\).

6. For all \(n \ge 1\), the \(f_n^{(i)}\) have a common primitive zero in \((\mathbb {Z}/ p^n \mathbb {Z}_p)^m\).

The Hilbert symbol statisfies the following formulas:

1. For \(a, b \in k\), we have \((a, b)_k = (b, a)_k\).

2. Let \(a, c \in k^\times \). Then \((a, c^2)_k = 1\).

3. For \(a \in k^\times \), we have \((a, -a)_k = 1\).

4. Let \(a \in k\) with \(a \ne 0, 1\). Then \((a, 1 - a)_k = 1\).

5. Let \(a, a', b \in k^\times \) and assume that \((a, b)_k = 1\). Then \((a a', b)_k = (a', b)_k\).

6. Let \(a, b \in k\). Then \((a, -ab)_k = (a, b)_k\).

7. Let \(a, b \in k^\times \) with \(a \ne 1\). Then \((a, (1-a)b)_k = (a, b)_k\).

Let \(R\) be a commutative ring. If a quadratic form \(Q\) on an \(R\)-module \(V\) represents \(0\) and is nondegenerate, then \(Q\) is equivalent to \(H + Q'\), where \(H\) is hyperbolic. Moreover, \(f\) represents all elements of \(R\).

Let \(a, b \in \mathbb {Q}^\times \). Then \((a, b)_p = 1\) for all but finitely many primes \(p\).

Let \(S\) be a finite set of prime numbers. The image of the finite embedding of \(\mathbb {Q}\) into \(\mathbb {R}\times \prod _{p \in S} \mathbb {Q}_p\) is dense with respect to the product topology. More concretely, for every \(\epsilon {\gt} 0\) and every \(y \in \mathbb {R}\times \prod _{p \in S} \mathbb {Q}_p\), there exists \(x \in \mathbb {Q}\) such that the distance between \(y\) and the image of \(x\) is less than \(\epsilon \).

The Hilbert Symbol is bilinear in both variables, i.e. for all \(a, a', b, b' \in k^\times \), we have

Let \((a_i)_{i \in I}\) be a finite family of nonzero rational numbers, and let \(e_{i,v} \in \{ \pm 1\} \) for each \(i \in I\) and each place \(v\) of \(\mathbb {Q}\). There exists a rational number \(x \in \mathbb {Q}^\times \) such that

if and only if the following conditions hold:

1. For each \(i \in I\), we have \(e_{i,v}=1\) for all but finitely many places \(v\).

2. For each \(i \in I\), we have \(\prod _v e_{i,v}=1\).

3. For each place \(v\), there exists \(x_v \in \mathbb {Q}_v^\times \) such that \((x_v, a_i)_v = e_{i,v}\) for all \(i \in I\).

Let \(V\) be a \(K\)-vector space of dimension at least \(3\), let \(Q\) be a nondegenerate quadratic form on \(V\), and let \(b, b'\) be two \(Q\)-orthogonal bases of \(V\). Then there exists a finite sequence \((b^{(0)}, \cdots , b^{(m)})\) of orthogonal bases of \(V\) such that \(b^{(0)} = b\), \(b^{(m)} = b'\) and \(b^{(i)}\) is contiguous with \(b^{(i + 1)}\) for \(0 \le i {\lt} m\).

Let \(f\) be a quadratic form over \(\mathbb {Q}\). Then \(f\) is isotropic if and only if \(f\) is everywhere locally isotropic. In other words, \(f\) has a global zero if and only if \(f\) has everywhere a local zero.

Let \(x,y \in (\mathbb {Q}_2)^\times \). Then

Let \(p\) be an odd prime and let \(a, b \in (\mathbb {Q}_p)^\times \). Then

where \(\left(\dfrac {\cdot }{p}\right)\) is the Legendre symbol.

\({\mathbb {Q}_p^\times }^2\) is an open subgroup of \(\mathbb {Q}_p^\times \).

Let \(a, b \in \mathbb {Q}^\times \). Then \(\prod _{v \in V} (a, b)_v = 1\).

Let \(a, b \in \mathbb {R}^\times \). Then \((a, b)_{\mathbb {R}} = 1\) if and only if \(a {\gt} 0\) or \(b {\gt} 0\); otherwise \((a, b)_{\mathbb {R}} = -1\).