**Lecture 1**: Philosophy and basic notions

**Lecture 2**: Finite field extensions

**Lecture 3**: Algebraic closures

**Lecture 4**: Galois extensions

**Lecture 5**: Transcendence bases, and Pell's equation I

**Lecture 6**: Extensions and the revenge of Pell's equation

**Lecture 7**: Profinite Galois theory

**Lecture 8**: Galois Cohomology

**Lecture 9**: The idea of the Brauer-Manin obstruction

**Lecture 10**: Brauer-Manin obstructions, part 2: the structure of the Brauer group

**Lecture 11**: Brauer-Manin obstructions, part 3: computing the Brauer group of varieties