1
vote
1answer
24 views

How do i find a signature of a quadratic form? Also how do i represent a quadratic form as a sum/difference of squares?

For example given $(x,y,z,t) = xy+ y^2+ yz+z^2+zt$ How do i represent it as a sum and difference of squares (i.e. in the form $\sum a_iA_i^2$) and how do i find its trace? Or if i have a quadratic ...
2
votes
0answers
49 views

Connection between class number and the theory of Ideals/Quadratic Fields

I've been studying the classic results in integer binary quadratic forms, mainly the equivalence and reduction of quadratic forms and the class number $H(d)$ (the definition I got for $H(d)$ is the ...
1
vote
0answers
76 views

Does this quadratic form represent 1?

I am stuck on the following question in Lam's quadratic forms for a few days now. Let $a,b,c$ be three elements of a field $F$ such that $0 \neq a^2+b^2 \neq c^2$. Show that the quadratic form ...
1
vote
1answer
48 views

Showing that the quadratic form $Q(x,y,z,t)=x^2+y^2+z^2-7\cdot t^2$ is anisotropic on $\Bbb Q^4$

I'm looking for help in order to find a prove that the quadratic form $Q(x,y,z,t)=x^2+y^2+z^2-7\cdot t^2$ on $\mathbb Q^4$ can or cannot take the value $0$ on a nonzero element of $\Bbb Q^4$. I was ...
3
votes
2answers
76 views

Where does the theory of quadratic forms fail in characteristic 2?

Let $V$ be a finite-dimensional vector space over a field $k$, and $Q$ a nondegenerate quadratic form on $V$. If the characteristic of $k$ is not 2, then we can change coördinates on $V$ so that ...
6
votes
2answers
107 views

Question on quadratic forms

I know a theorem which says: If a non-singular quadratic form (homogeneous polynomials of degree $2$) over a field $K$ represents zero non-trivially (i.e., there is a nontrivial solution of the ...
3
votes
1answer
90 views

Question about the definition of representability of a quadratic form

Say I have an integral quadratic form $q$ on $\mathbb{Z}^r$ and another integral binary quadratic form $\tilde{q}$. What does it mean for $q$ to primitively represent $\tilde{q}$? I can't seem to find ...
0
votes
3answers
118 views

$n$-linear alternating form with $\dim{V}<n$ $\overset{?}{\text{is}}$ the $0$-form

Prove that every $n$-linear alternating form on a vector space of dimension less than $n$ is the zero form.
1
vote
1answer
50 views

$n$-linear form: An Interpretation

What is a good example of an $n$-linear form that is more familiar to a student learning at an elementary level? EDIT: I'm just trying to show that every $n$-linear alternating form on a vector ...
2
votes
1answer
128 views

Simultaneous Orthogonalization

Let $q,q':\mathbb V \longrightarrow \mathbb R$ be two quadratic form where $\mathbb V$ is vector space with $dim \mathbb V \geq3$ and $q(x)+q'(x)>0$ for any $0\neq x\in \mathbb V$ then there exists ...
1
vote
2answers
226 views

Isometry without injection and surjection

Suppose that $B_1$ and $B_2$ are bilinear form on space $V_1$ and $V_2$. An isometry relative to $B_1$ and $B_2$ is an linear map $\sigma:V_1 \rightarrow V_2$ satisfying ...
0
votes
1answer
88 views

A quadratic space over an algebraically closed field is isotropic

Let $F$ be an algebraically closed field, and let $(V,f)$ be a quadratic space over $F$. How can one show that if $\dim V \geq 2$ then it is an isotropic space? Thanks.
7
votes
2answers
182 views

How does the theory of the quadratic number fields relate to the quadratic forms?

As every one knows, the quadratic number fields shares a deep connection with the binary quadratic forms; I have been told this relation when I was a senior high, and now I, learning some difficult ...