= Independent quaternion classes over rational numbers
{title2=$\bigoplus_{p\equiv3\ (4)}\mathbb Z/2\hookrightarrow\operatorname{Br}(\mathbb Q)$}
Associate to each prime $p\equiv3\pmod4$ the class of $(-1,p)_{\mathbb Q}$. These classes have order two, since quaternion conjugation identifies the algebra with its opposite. The product rule $[(-1,a)][(-1,b)]=[(-1,ab)]$ follows by factoring their <tensor product> into the commuting subalgebras generated by $i_1,j_1j_2$ and by $i_1i_2,j_2$; the second is the split algebra $(1,b)$. Every nonempty finite product of distinct such primes has an odd valuation at one of them. The <quaternion norm obstruction at primes congruent to three modulo four> therefore makes its class nonzero, proving independence. Euclid's construction $4p_1\cdots p_m-1$ proves there are infinitely many such primes.
Back to article page
