# Nepal Algebra Project (NAP) / नेपाल बीज-गणित परियोजना # COURSE YEAR 2016

Module - V : July 10, 2016 – July 22, 2016, Kalyan Chakraborty
Topics: Applications of Galois Theory: Primitive element theorem, Funda-mental Theorem of Algebra, Cyclotomic extensions, Dedekind’s theorem on the independence of characters, Normal basis theorem, Hilbert’s Theorem 90, Cyclic extensions, Kummer theory, Proof of Galois’s solvability theorem, Symmetric polynomials, General polynomial of degree n, Norms and traces

• Recalled separable' criterion with examples and also recalled the Fundamental theorem of Galois theory'. Then motivated the Primitive element theorem' with an example and gave the proof of the theorem. Then I showed that if $E$ is a separable, algebraic extension of $F$ then there can be only finitely many intermediate fields. Also proved the converse of this result and in particular mentioned that in case of Galois this converse provides another proof of the Primitive element theorem'.

• Proved The fundamental theorem of algebra' using Galois theory. Then recalled basic facts about $n$-th roots of unity and in particular make them familiar with primitive $n$-th roots of unity. Then introduced the cyclotomic extensions' with fair bit of motivation. Then proved the main result about these extensions by showing that if $F$ is a field with either characteristic $0$ or a prime $p$ with $p$ not dividing a given positive integer $n$ and $E = F[\zeta]$ with $\zeta$ a primitive $n$-th root of unity then this extension is a Galois extension and the Galois group is embedded into the abelian group $(\mathbb{Z}/n\mathbb{Z})^*$ Concluded with an example showing that this embedding need not be surjective.

• Introduced cyclotomic polynomials with examples. Then Continuing from the last lecture I showed that the injective homomorphism $Gal \left( F[\zeta]/F \right)$ $\longrightarrow \left({\mathbb Z}/n{\mathbb Z}\right)^*$ is in fact an isomorphism incase when $F = {\mathbb Q}$ by showing that the $n$-th cyclotomic polynomial $\Phi _n$ is irreducible over ${\mathbb Q}[x]$. Then after introducing characters I proved the Dedekind's theorem on linear independence of characters' and re-stated it in the format of Galois extensions as a corollary.

• Normal basis in case of a finite Galois extension was defined with some interesting examples. Then proved the normal basis theorem' as an application of Dedekind's theorem on independence of characters'. The theorem was proved in case of infinite fields with some discussion in the case of finite fields.
• Crossed homomorphisms and principal crossed homomorphisms were defined and $H^1(G,M)$, the first cohomology group where $M$ is a $G$-module, was introduced. Then it was shown that in case of finite Galois extensions every crossed homomorphisms are in fact principal crossed homomorphisms, i.e. , $H^1(G, E^*)$, where $G$ is the Galois group of an extension $E$ over $F$, is in fact trivial. This was proved again as an application of the Dedekind's theorem on independence of characters. In the next lecture Hilbert Theorem 90' will be deduced as a corollary of this result.

• Proved Hilbert's theorem 90' and deduced the parametrisation of the 'Pythagorean triplets' as an application of this theorem.
• We discussed the cyclic extensions of general degree $n$ when the base field contains a primitive $n$-th root of unity. Then the characterisation of such extensions were shown.
• Concluded with an example of a $5$-th degree polynomial showing that it is not solvable as its Galois group is not solvable.