Hello expensive customer to our community We will proffer you an answer to this query reference request – Explicit isomorphism for quaternion algebras over $mathbb{Q}$? ,and the respond will breathe typical by means of documented data sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning concerning the respond to this query.

reference request – Explicit isomorphism for quaternion algebras over $mathbb{Q}$?

Timo Hanke has written on this drawback within the generality of cyclic algebras (http://arxiv.org/abs/math/0702681). He reveals that it’s equal to the answer of a norm equation, which has an algorithmic answer over world fields. In your illustration, this could in common contain the answer of a norm equation over a *biquadratic* bailiwick; I do not know the way nicely this could fare in drill.

In the particular illustration of quaternion algebras that you’re inquiring about, there’s a hyperlink to quadratic varieties, and as Keith Conrad mentions, it’s equal to discover a zero of a quadratic figure in six variables. This goes advocate to Albert, who checked out this figure intimately in organize to show that there was an “honest” (non-quaternion) biquaternion algebra. reference for that is part 16 of the “Book of Involutions” or part XII.2 of Lam’s “Introduction to Quadratic Forms over Fields”. It might appear love only a reformulation of the issue, but it surely turns on the market are algorithmic strategies to seek out factors on quadrics over quantity fields which are fairly environment friendly in drill: the buzzword right here is “indefinite LLL”, and Watkins (http://magma.maths.usyd.edu.au/~watkins/papers/illl.pdf) explains what Magma does to achieve this job.

In common, right here is an thought I kicked round as soon as which a minimum of reduces the issue to decipher norm equations over quadratic extensions (as a substitute of biquadratic extensions). This energy breathe solely of theoretic/algorithmic curiosity, however a minimum of it probably generalizes. We want to check if $A cong B$ over a world bailiwick $F$ (say of attribute not $2$ for now), and if that’s the case, to seek out an categorical isomorphism. If $A=(a,b)$ and $B=(c,d)$ and $a=c$, then there may be an isomorphism if and provided that $b/d$ is a norm from $mathbb{Q}(sqrt{a})$, and this will breathe completed algorithmically by a norm equation over this bailiwick; so it is sufficient to dwindle to this illustration. To discover a frequent subfield $Ok=mathbb{Q}(sqrt{a})$ in $A,B$, one can merely choose one (select $Ok$ such that $K_v$ is just not splinter in any respect locations $v$ ramified in $A$ and $B$, e.g., take $a=-mathrm{lcm}(ab,cd)$ if $gcd(a,b)=gcd(c,d)=1$, so this step doesn’t plane require factoring). The drawback of embedding a quadratic bailiwick $Ok$ in a quaternion algebra is equal to (checking if $Ok$ splits the algebra and so) to a norm equation (a benchmark outcome, behold e.g. http://www.math.dartmouth.edu/~jvoight/articles/quatalgs-060513.pdf). Once the bailiwick is embedded, we will diagonalize the quadratic figure to dwindle to the illustration the place $a=c$.

This method most likely generalizes to cyclic algebras (of any attribute), and whether it is fascinating to you, it’s one thing I’d breathe fortunate to labor out with you.

we’ll proffer you the answer to reference request – Explicit isomorphism for quaternion algebras over $mathbb{Q}$? query through our community which brings all of the solutions from a number of dependable sources.

## Add comment