All Questions

Consider the field of fractions $K$ of the quotient algebra $\mathbb{R}[x,y,z,t]/(x^2+y^2+z^2+t^2+1)$, where $\mathbb{R}$ is the field of real numbers and $x,y,z,t$ are variables. Clearly $-1$ is a ...

Associated to a finite, separable field extension $L/K$, there is a natural nondegenerate bilinear form, the trace form, defined by $$\langle x,y \rangle := \mathrm{Tr}_{L/K}(xy)$$ Now, given a ...

I found this topic in a book 'Metric Affine Geometry' by Ernst Snapper and Robert J. Troyer. I call a field $k$ trigonometric iff there is a quadratic form $q$ over $k^2$ such that every two lines ...

I have spent the past two years studying abstract Witt rings. These objects are a generalization of "The Witt ring of a field," an algebraic invariant of fields of characteristic not equal to 2. ...

Let $\ell$ be a division ring, and let $k$ be a sub division ring. I know that a quadratic equation $x^2 + ax + b = 0$, with $a, b \in k$ can have more than two solutions in $\ell$, but what if the ...

Let $\ell$ be a skew field (i.e., a division ring), and let $k$ be a sub skew field, such that the dimension of $\ell$ as a left vector space over $k$ is $2$. Then if $a \in \ell \setminus k$, we can ...