In the first part of the seminar we will quickly review the theory of symmetric bilinear forms over fields. We will introduce the Grothendieck-Witt ring $GW(K)$ and the Witt ring $W(K)$ of a field $K$ and make some basic examples, e.g. the cases where $K$ is algebraically closed or real closed.
The second part of the seminar will be directed to the study of the Witt ring of the field of rational numbers. The structure of $W(\mathbb Q)$ will be deduced from the structure of $W(\mathbb Q_p)$ and $W(\mathbb R)$ via the Hasse-Minkowski theorem. In order to have a formula for $W(\mathbb Q_p)$ we will need to work a bit with the properties of squares in $\mathbb Q_p$ as well as with $W(\mathbb F_p)$.
In the third part of the seminar we will see some basics on Pfister forms. If time permits we will apply the results we have seen during the seminar to some classical elementary question such as: