[Pie 1.6], [MR 2.1, up to Lemma 2.1.6]
Introduce quaternion algebras $A = (\frac{a,b}{F})$; mention that they are (the simplest non-trivial examples of) central simple algebras. If $A$ is a division algebra, then there is a quadratic field extension $E \subset A$ of $F$ [MR, Lemma 2.1.6].

[MR 2.8]
Introduce central simple algebras and the Brauer-group

[MR 2.9]
Prove Wedderburn's structure Theorem [MR Thm 2.9.6] and the Skolem Noether Theorem [MR Thm 2.9.8] [Pie 12.6]. As an example application, show [MR, Thm 2.1.7] and [MR, Thm 2.1.8].

[MR 2.3] [Pie 1.7]
Express when two quaternion algebras are isomorphic in terms of quadratic forms [MR Cor. 2.3.5] [Pie 1.7 Prop] As an example, classify all quaternion algebras over ℝ (see also [MR Thm 2.5.1])

[Pie 13.1-13.3]; but restrict to the case where $A$ is a central simple algebra
[Pie 13.1]: Introduce the notion of (strictly) maximal subfield of a division algebra. Introduce the notion of degree (see [Pie 13.1], after Cor. a). Prove that in division algebra, maximal subfields are strictly maximal [Pie 13.1 Cor b]. Skip [Pie 13.1 Thm] (this has been done in a previous talk), but deduce that the Brauer group of $\mathbb{R}$ is $\mathbb{Z}/2\mathbb{Z}$. [Pie 13.2]: Introduce the notion of splitting field. Introduce the relative Brauer group $\mathbf{B}(E/F)$. Show that if $E$ is a maximal subfield of a division algebra $A$, then $A$ splits over $A$ [Pie 13.3 Theorem]

Part a: [Pie 13.5]
Prove that the Brauer group $\mathbf{B}(F)$ is the union of the $\mathbf{B}(E/F)$ for all Galois extensions $E$ of $F$ [Pie 13.5 Corollary]. (Or, if the proof is boring, just sketch it)
Part b: [Pie 13.6]
Let's have some fun now: Prove Wedderburns Finite Division Algebra Theorem [Pie 13.6] (stating that there are only fields)

[Pie 14.1-14.3]
Sketch the proof that $\mathbf{B}(E/F)$ is isomorphic to $H^2(\operatorname{Gal}(E/F), E^\times)$ [Pie 14.2 Theorem]. In particular, introduce the crossed product $[(E, G, \Phi)]$ [Pie 14.1 Corollary].

[Pie 15.1 15.4]
Introduce cyclic algebras and the notation $(E, \sigma, a)$ ([Pie 15.1, after Prop. a]); prove at least [Pie 15.1 Cor. a (i)] and [Pie 15.1 Prop b]
Present the example construction $(\frac{a,b}{F,\zeta})$ from [Pie 15.4] (if there is time left), as a generalization of quaternion algebras.

[Pie 17.1-17.3]
Introduce the following notions for division algebras: valuations [Pie 17.1] (maybe give the example from Exercise 3 (c)); non-archimedean valuations [Pie 17.2]; valuation rings [Pie 17.3]
All this serves as an excuse to recall the corresponding material on fields.

[Pie 17.4, 17.5]
Introduce the valuation topology, completion, and Hensel's lemma [Pie 17.4]. Introduce local fields [e.g. Pie 17.5]

Part a: [MR 2.6]
Classify quaternion algebras over non-archimedean local fields [MR 2.6]. (You can also restrict to $\mathbb{Q}_p$, if you want. Don't do all details if it would take too much time.)
Part b: [Pie 17.6]
Prove that the valuation on a local field $F$ extends uniquely to any finite dimensional division algebra over $F$ [Pie 17.6 Prop].

[Pie 17.7, 17.8]
Introduce the notion of ramification and prove the formulas from [Pie 17.7 Prop]. Describe unramified extensions of non-archimedean local fields [Pie 17.8 Proposition].

[Pie 17.9, 17.10]
Recall why we are interested in the norm factor group $F^\times/\mathrm{N}(K^\times)$ (namely, because of [Pie 15.1 Prop b]) and determine that group for an unramified extension $K$ of a local field $F$ [Pie 17.9 Prop].
Put things together to determine the Brauer group of non-archimedean local fields [Pie 17.10 Thm].