Brauer group of a field of rational numbers Can we say anything about Brauer group of $\mathbb{Q}$? And how can we construct it?
 A: The Brauer group of $\mathbb{Q}$ can be identified with a subgroup of $\bigoplus_vBr(\mathbb{Q}_v)$, the direct sum taken over all places of $\mathbb{Q}$ (including the Archimedean one). The map from $Br(\mathbb{Q})$ into this direct sum is given by tensoring any central simple algebra representing a class of the Brauer group with all the completions, to get central simple algebras over the completions. This tensor product will be a split CSA for almost all $v$, so you do land in the direct sum, rather than in the direct product.
Each of the local Brauer groups at the finite places is isomorphic to $\mathbb{Q}/\mathbb{Z}$, while $Br(\mathbb{R})\cong\frac12\mathbb{Z}/\mathbb{Z}$, the isomorphism given by the Hasse invariant. From the direct sum, you have a map to $\mathbb{Q}/\mathbb{Z}$ that simply adds the Hasse invariants, and the Brauer group of $\mathbb{Q}$ in the direct sum is precisely the kernel of this map.
A source for all of this would be, for example, the book on algebraic number theory edited by Cassels and Froehlich, specifically the chapters on local and global class field theory.
A: The Brauer group of $\mathbb Q$ (and more generally, of any finite extension of $\mathbb Q$) is described by class field theory.
The answer is that it is isomorphic to the direct sum of $\mathbb Z/2 \mathbb Z$
and  a countable number of copies of $\mathbb Q/\mathbb Z$.
More canonically, consider the direct sum
$$\dfrac{1}{2}\mathbb Z/\mathbb Z \oplus \bigoplus_p \mathbb Q/\mathbb Z,$$
where $\dfrac{1}{2}\mathbb Z/\mathbb Z$ denotes the (unique) cyclic
subgroup of order $2$ in $\mathbb Q/\mathbb Z$, and the direct sum is indexed by primes.
There is a natural map from this direct sum to $\mathbb Q/\mathbb Z$, given
by summing components, and the Brauer group of $\mathbb Q$ is canonically identified with its kernel.
The elements of order $2$ correspond to quaternion algebras over $\mathbb Q$,
and these are constructed explicitly in Serre's book A course in arithmetic.
Off the top of my head I'm not sure about computing them explicitly more generally.  
[Note: I wrote this answer several hours ago, but just got around to submitting it now.  Thus it likely duplicates the other answers.]
A: $\DeclareMathOperator{\br}{Br}$
This can be done using the Brauer-Hasse-Noether theorem. One statement of this theorem is that for $k$ a global field, there is a canonical exact sequence 
$$
  0 \to \br(k) \to \bigoplus_v \br(k_v) \to \mathbb Q/\mathbb Z \to 0 .
$$
where $v$ ranges over all places (finite and infinite) of $k$. It is known (see for example Serre's Local Fields, chapter 13) that $\br(k_v)\simeq \mathbb Q/\mathbb Z$. So for $k=\mathbb Q$, our sequence is 
$$
  0 \to \br(\mathbb Q) \to \mathbb Z/2\times \bigoplus_p \mathbb Q/\mathbb Z \to \mathbb Q/\mathbb Z\to 0.
$$
So we have 
$$
  \br(\mathbb Q) = \left\{(a,x):a\in \{0,1/2\}\text{ , }x\in \bigoplus_p \mathbb Q/\mathbb Z\text{ and }a+\sum x_p=0\right\}.
$$
We have similar descriptions of $\br(k)$ for any global field $k$. 
