Modern Algebra (ECM3731) Part 2: Rings, fields and Galois theory

The departmental page for Modern Algebra (ECM3731).
The ELE page for Modern Algebra (ECM3731).

Remarks about the course

Learning outcomes

Remarks about the lectures

Final remark about finding the Galois group of a finite normal extension $F\subseteq K$ in $\mathbb C$.
If $K$ is given as $K=F(\alpha_1,...,\alpha_n)$, then any $F$-linear automorphism is determined by what it does to the $\alpha_i$. Furthermore, an element of $K$ can only be moved to one of its conjugates over $F$. So if $\alpha_i$ has $m_i$ conjugates over $F$ (including itself), then that gives us $m_1\cdots m_n$ possible automorphisms. Just choose for each $\alpha_i$ a conjugate over $F$. If you can somehow show that the order of the Galois group is also $m_1\cdots m_n$ (usually by showing that $[K:F]=m_1\cdots m_n$), then you know that all above possibilities must occur and you have determined the Galois group as a set. Then you can pick out suitable automorphisms that move one of the $\alpha_i$ and fix all the others, and show that they generate the Galois group. This gives the Galois group as a set and in small examples you can determine the group structure by determining a commutation rule for the generators. The above cardinality argument avoids the application of the argument from the proof of Thm III.5 that I used in the notes and the solutions to the exercises. The latter is more insightful, but the former is shorter.