TITLE: Explicit description of a quaternion algebra with a prescribed set of ramified places
QUESTION [8 upvotes]: Let $k$ be an algebraic number field. I understand that given a finite set of non-complex places $S\subset V(k)$ of even cardinality, there exists a unique quaternion algebra $Q$ over $k$ such that $Q$ ramifies at all $v\in S$ (i.e. $Q \otimes_k k_v$ is a division algebra) and $Q$ splits at all the other places. Is there a way to get an explicit description of $Q$ of the form $Q(a,b|k)$ for some parameters $a,b, \in k^{\times}$?
The reason for my question is to find an explicit example of a division quaternion algebra over the field $\mathbb{Q}(\sqrt[4]{2})$ that splits at all the archimedean places and is moreover of the form $Q_0 \otimes_{\mathbb{Q}(\sqrt{2})} \mathbb{Q}(\sqrt[4]{2})$ for a quaternion algebra $Q_0$ defined over $\mathbb{Q}(\sqrt{2})$.
Any hint on the general question or on the specific example is greatly appreciated. Thank you!

REPLY [12 votes]: I wrote something that makes this process algorithmic (and tries to simplifying the resulting algebra as much as possible) in Magma.
Copy and paste the following code into Magma (e.g. the calculator at http://magma.maths.usyd.edu.au/calc/)
_<x> := PolynomialRing(Rationals());  
F<s> := NumberField(x^2-2);  // s = sqrt(2)  
_<xF> := PolynomialRing(F);  
K<w> := ext<F | xF^2 - s>;  // w = 2^(1/4) = sqrt(s)  
ZK := Integers(K);  
// find small primes that are not split in K  
pps := [pp : pp in PrimesUpTo(20,F) | #Factorization(ZK!!pp) eq 1];  
// Compute a quaternion algebra (over F) ramified at 2 finite primes 
// and no infinite places  
B := QuaternionAlgebra(&*pps[1..2]);   
B;  
// Verify ramification  
RamifiedPlaces(B);  
RamifiedPlaces(ChangeRing(B, AbsoluteField(K)));  

The algorithm it uses is probabilistic, since there is no clear "best" quaternion algebra with specified ramification set (and anyway may be expensive to compute).  Anyway, on this run it tells me
Quaternion Algebra with base ring F, defined by i^2 = -s - 1, j^2 = 8*s + 11

So you can take $i^2 = a = -\sqrt{2}-1$ and $j^2 = b = 8\sqrt{2}+11$; these are elements of smaller norm than you could get from $\mathbb{Q}$.