$\DeclareMathOperator{\Q}{Q} \DeclareMathOperator{\Aut}{Aut}$

# COURSE YEAR 2018

Module - III : June 03, 2018 – June 15, 2018, Michel Waldschmidt
Topics: The fundamental theorem of Galois theory (FTGT), Examples and applications of FTGT, Constructions with straight-edge and compass, The Galois group of a polynomial, Solvability of equations.
[Homeworks]

• Introduction to Galois work. Solving equations by radicals. Fields, extensions; subfields. Question: which are the subfields of $\mathbb{Q}$?
• Characteristic of a field. Field with $4$ elements. Construction by hand, next as stem field of $X^2+X+1$ over $\mathbb{F}_2$.
• Proof of the fact that the number of elements of a finite field is $p^r$ where $p$ is the characteristic and $r$ the dimension of the field as a vector space over the prime field.

• Finite extensions: definition, example of finite extensions, examples of extensions which are not finite.
• Separable extensions: definition, example of separable extensions, examples of extensions which are not separable. Case of characteristic zero. Case of finite fields.
• Normal extensions: definition, example of norma extensions; quadratic extensions. Examples of extensions which are not normal.
• Galois extensions; equivalent definitions.
• Construction of a field with $p^r$ elements ($p$ prime $r\ge 1$). Unicity with a unique isomorphism for $r=1$. Automorphisms of a finite field: cyclic group of order $r$ generated by the Frobenius.

--What is the difference between an extension and a splitting field ?
--In an extension, is it true that the two fields have the same characteristic?
• Erratum to Remark 3.11 (b) p.~38 in Milne's notes (pointed out by Roger Wiegand). Corrected version, following Hironori Shiga, NAP 2017 Module III Problem 1
http://www.rnta.eu/nap/nap-2017/course-2017/module_3_hw_1.pdf

Let $E/F$ be a separable finite extension and let $G$ be a finite subgroup of $\mathrm{Aut}(E/F)$. Then Proposition 2.7 (a) says $\#\mathrm{Aut}(E/E^G)$ $=[E:E^G]$. On the other hand, by (3.5), we have $\mathrm{Gal}(E/E^G)$ $=G=$ $\mathrm{Aut}(E/E^G)$, therefore $\# G=$ $[E:E^G]$.
In particular, when $E/F$ is Galois with Galois group $G$, we have

$G=$ $\mathrm{Gal}(E/F)$ $=$ $\mathrm{Aut}(E/F)$ and $\# \mathrm{Gal}(E/F)$ $=$ $[E:F]$.

• For an extension $E/F$ where $E=$ $F(\alpha_1,$ $\dots,$ $\alpha_m)$, an element of $\mathrm{Aut}(E/F)$ is determined by its values at $\alpha_1,$ $\dots,$ $\alpha_m$.
In particular, for a simple extension $E=F(\alpha)$, an element of $\mathrm{Aut}(E/F)$ is determined by its values at $\alpha$.
Conjugates of an algebraic element over $F$ (roots of the irreducible polynomial). If $\alpha$ has $r$ conjugates over $F$ in $F(\alpha)$, then $\mathrm{Aut}(F(\alpha)/F)$ has at most $r$ elements.
For a Galois extension $E/F$ of degree $d$, $\mathrm{Aut}(E/F)$ is a group of order $d$.
• Statement of the Fundamental Theorem of Galois Theory (Theorem 3.16).

-- Is there a connection between ideals and classes ?
• Background on algebra: groups, subgroups, classes, normal subgroups, quotient of a group by a normal subgroup, canonical surjection.
Commutative rings, ideals, quotient. Examples: $\mathbb{Z}$, $F[X]$. When is the quotient a field in these two examples?
Cyclic groups, subgroups, quotient. Product of cyclic groups.
Groups of order a prime number. Groups of order $\le 6$.
• Corollary 3.12: if $E/F$ is separable there is a finite extension of $E$ which is Galois over $F$.
• Corollary 3.13. Let $E\supset$ $M\supset$ $F$ be finite extensions. If $E/F$ is Galois, then $E/M$ is Galois. Examples where
-- $E/F$ is Galois and $M/F$ is not Galois
-- $E/M$ and $M/F$ are Galois and $E/F$ is not Galois.
• The splitting field $E$ of $X^3-2$ over $\mathbb{Q}$: $\mathrm{Gal}(E/\mathbb{Q})$ as a permutation group of the roots.

--- In a ring $A$ (commutative with unity), the group of units $A^\times$, irreducible elements: definition, examples: fields, $\mathbb{Z}$, $K[X]$, $\mathbb{Z}[X]$. Irreducible elements in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$.
--- Order of an element $x$ in a group $G$: kernel of the homomorphism $\mathbb{Z}\to G$ which maps $n$ to $x^n$.
--- Cyclic groups. Subgroups. Direct products of cyclic groups. Finite groups of order $\le 7$ (already done in fourth course -- it seemed necessary to repeat).
--- The group $\mathrm{GL}_2(\mathbb{Q})$.
--- The quartic extension $E=$ $\mathbb{Q}(i,\sqrt{2})$ of $\mathbb{Q}$. Galois group, subgroups, subfields, Galois correspondence between subfields of $E$ and subgroups of the Galois group.
--- Given a Galois extension $E/F$ and a subfield $M$ of $E$ containing $F$, necessary and sufficient condition for $M$ to be Galois over $F$.
• The Galois group of a separable polynomial of degree $n$ as a subgroup of $\mathfrak{S}_n$. Transitive subgroups of $\mathfrak{S}_n$. The Galois group of $f$ over $F$ is transitive if and only if $f$ is irreducible over $F$. Example: the Galois group of $(X^2-2)$ $(X^2+1)$ over $\mathbb{Q}$.

--- In a ring $A$ (commutative with unity), the group of units $A^\times$, irreducible elements: definition, examples: fields, $\mathbb{Z}$, $K[X]$, $\mathbb{Z}[X]$. Irreducible elements in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$.
--- Order of an element $x$ in a group $G$: kernel of the homomorphism $\mathbb{Z}\to G$ which maps $n$ to $x^n$.
--- Cyclic groups. Subgroups. Direct products of cyclic groups. Finite groups of order $\le 7$ (already done in fourth course -- it seemed necessary to repeat).
--- The group $\mathrm{GL}_2(\mathbb{Q})$.
--- The quartic extension $E=$ $\mathbb{Q}(i,\sqrt{2})$ of $\mathbb{Q}$. Galois group, subgroups, subfields, Galois correspondence between subfields of $E$ and subgroups of the Galois group.
--- Given a Galois extension $E/F$ and a subfield $M$ of $E$ containing $F$, necessary and sufficient condition for $M$ to be Galois over $F$.
• The Galois group of a separable polynomial of degree $n$ as a subgroup of $\mathfrak{S}_n$. Transitive subgroups of $\mathfrak{S}_n$. The Galois group of $f$ over $F$ is transitive if and only if $f$ is irreducible over $F$. Example: the Galois group of $(X^2-2)$ $(X^2+1)$ over $\mathbb{Q}$.

• Answer to questions dealing with Problem Set 1.
• Constructions with straight edge and compass (Milne p. 20-23)
• Definitions. Examples: the set $\mathcal{C}$ of constructible numbers contains $\mathbb{Q}$. Construction of $\sqrt{c}$ for $c\in\mathcal{C}$.
• Theorem 1.36: the set of constructible numbers is a field; a number $\alpha$ is constructible if and only if it belongs to a subfield of $\mathbb{R}$ of the form $\mathbb{Q}$ $(\sqrt{a_1},\sqrt{a_2},$ $\dots,\sqrt{a_r})$ where $a_1\in\mathbb{Q}$ and for $i=2,$$\dots,r$, $a_i\in\mathbb{Q}$ $(\sqrt{a_1},\sqrt{a_2},$ $\dots,\sqrt{a_{i-1}})$.
• The field obtained by adjoining to $\mathbb{Q}$ the square roots of prime numbers is the same as the field obtained by adjoining all the square roots of positive rational numbers ( It took some time to convince the students that the two fields are the same ).
• Example of a constructible number: $\sqrt{(\sqrt 2 +2)\sqrt 3 +3)}$ (cf. Exercise 3.4).
• Corollary 1.37: constructible numbers are algebraic numbers with degree a power of $2$.
• Corollary 1.38: duplication of the cube.
• Corollary 1.39: trisection of the angle; $\cos (\pi/9)$ is not constructible.
• Construction of regular polygons. Fermat numbers, Fermat primes.

Theorem 3.23. If a positive real number is contained in a subfield of $\mathbb{R}$ which is a Galois extension of $\mathbb{Q}$ of degree a power of $2$, then it is constructible.
Uses the fact that a group of order a power of $2$ is solvable.
Corollary: for $p$ a Fermat prime $p=$ $F_n=$ $2^{2^n}+1$, a regular polygon with $p$ sides can be constructed with ruler and compass.
The Galois group of $\mathbb{Q}(e^{2i\pi/p}$ over $\mathbb{Q}$. The group $(\mathbb{Z}/p\mathbb{Z})^\times$ is cyclic. Example: $p=7$.
• Example of a constructible number: $\sqrt{(\sqrt 2 +2)(\sqrt 3 +3)}$ (Milne exercise 3.4 p. 46 and solution p.129).
In the solution, Milne implicitly defines $\alpha=\sqrt{(\sqrt 2 +2)\sqrt 3 +3)}$. When he considers $\sigma(\alpha^2)$ in the solution of (b), he uses the fact that $\alpha^2\in M$. Next he writes: Extend $\sigma$ to an automorphism of $E$. The following result should be added after Theorem 3.16.
Under the hypotheses of Theorem 3.16 (d), when $H$ is a normal subgroup of $G$ and $M=E^H$, the map $G$ $=$ $\mathrm{Gal}$ $(E/F)$ $\to$ $G/H$ $=$ $\mathrm{Gal}$ $(M/F)$ is a surjective homomorphism, hence for any $\tau$ $\in$ $\mathrm{Gal}$ $(M/F)$ there exists $\sigma$ $\in$ $\mathrm{Gal}$ $(E/F)$ such that the restriction of $\sigma$ to $M$ is $\tau$.