Nepal Algebra Project (NAP)

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.

Monday, June 04, 2018 (also download from here)

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.

Tuesday, June 5, 2018 (also download from here)

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.

Wednesday, June 6, 2018 (also download from here)

Answer to questions:
--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).

Thursday, June 7, 2018 (also download from here)

Answer to question:
-- 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.

Monday, June 11, 2018 (also download from here)

Answer to question:
--- 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 $\mathbb{Q}$.

Tuesday, June 12, 2018 (also download from here)

Answer to question:
--- 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 $\mathbb{Q}$.

Wednesday, June 13, 2018 (also download from here)

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.

Thursday, June 14, 2018 (also download from here)

Fermat numbers. Mersenne primes.
Constructions with straight edge and compass revisited (Milne p. 43-44)
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 $\tau$.
Remark: Proposition 3.18, 3;19 and 3.20 have hardly been discussed in the class for lack of time.

Homeworks: problem set 1, problem set 2

Solutions: problem set 1 , problem set 2

Nepal Algebra Project (NAP) / नेपाल बीज-गणित परियोजना

COURSE YEAR 2018