@ -6,6 +6,7 @@
\usepackage { enumerate}
\usepackage { hyperref}
\usepackage { amssymb}
\usepackage { tikz} % diagram
\begin { filecontents} [overwrite]{ galois-theory-notes.bib}
@misc{ ianstewart,
@ -14,9 +15,24 @@
year = { 2004}
}
@misc{ milneFT,
author={ Milne, James S.} ,
title={ Fields and Galois Theory (v5.10)} ,
year={ 2022} ,
note={ Available at \url { https://jmilne.org/math/} } ,
pages={ 144}
}
@misc{ berlekamp,
author={ Elmyn Berlekamp} ,
title={ Algebraic Coding Theory} ,
year={ 1984} ,
note={ Revised Edition from 1984}
}
@misc{ dihedral,
author = { Gaurab Bardhan and Palash Nath and Himangshu Chakraborty}
title = { Subgroups and normal subgroups of dihedral group up to isomorphism}
author = { Gaurab Bardhan and Palash Nath and Himangshu Chakraborty} ,
title = { Subgroups and normal subgroups of dihedral group up to isomorphism} ,
year = { 2010} ,
note = { \url { https://scipp.ucsc.edu/~haber/ph251/Dn_ subgroups.pdf} } ,
url = { https://scipp.ucsc.edu/~haber/ph251/Dn_ subgroups.pdf}
@ -62,7 +78,7 @@
\maketitle
\begin { abstract}
Notes taken while studying Galois Theory, mosty ly from Ian Stewart's book "Galois Theory" \cite { ianstewart} .
Notes taken while studying Galois Theory, mostly from Ian Stewart's book "Galois Theory" \cite { ianstewart} .
Usually while reading books and papers I take handwritten notes in a notebook, this document contains some of them re-written to $ LaTeX $ .
@ -158,7 +174,7 @@
From \ref { shorttowerlaw} .
\end { proof}
[...]
[...] TODO: pending to add key parts up to Chapter 15.
\newpage
@ -180,6 +196,9 @@ for $k=0, \ldots, n-1$.
So, by Euler's formula:
$$ z _ k = \sqrt [ n ] { r } \cdot e ^ { i ( \frac { \theta + 2 k \pi } { n } ) } $$
Usually we will set $ \alpha = \sqrt [ n ] { r } $ and $ \zeta = e ^ { \frac { 2 \pi i } { n } } $ ,
and find the $ \mathbb { Q } $ -automorphisms from there (see \ref { ex:galoisgroups} for examples).
\subsection { Einsenstein's Criterion} \label { einsenstein}
\emph { reference: Stewart's book}
@ -198,9 +217,110 @@ Then, $f$ is irreducible over $\mathbb{Q}$.
\emph { TODO from orange notebook, page 36}
\subsection { Cyclotomic polynomials} \label { cyclotomicpoly}
\emph { TODO theory from brown muji notebook, page 82}
\subsubsection { From Elmyn Berlekamp's "Algebraic Coding Theory" book}
The notes in this section are from the book "Algebraic Coding Theory" by Elmyn
Berlekamp \cite { berlekamp} .
\vspace { 0.3cm}
Some times we might find polynomials that have the shape of $ t ^ n - 1 $ , those are \emph { cyclotomic polynomials} , and have some properties that might be useful.
Observe that in a finite field of order $ q $ , factoring $ x ^ q - x $ gives
$$ x ^ q - x = x ( x ^ { q - 1 } - 1 ) $$
The factor $ x ^ { q - 1 } - 1 $ is a special case of $ x ^ n - 1 $ : if we assume that the
field contains an element $ \alpha $ of order $ n $ , then the roots of $ x ^ n - 1 = 0 $ are
$$ 1 , \alpha , \alpha ^ 2 , \alpha ^ 3 , \ldots , \alpha ^ { n - 1 } $$
and $ \deg ( x ^ n - 1 ) = n $ , thus $ x ^ n - 1 $ has at most $ n $ roots in any field, henceforth
the powers of $ \alpha $ must include all the $ n $ -th roots of unity.
There fore, in any field which contains a primitive $ n $ -th root of unity we have:
\begin { thm} { 4.31}
$$ x ^ n - 1 = \prod _ { i = 0 } ^ { n - 1 } ( x - \alpha ^ i ) = \prod _ { i = 1 } ^ n ( x - \alpha ^ i ) $$
\end { thm}
If $ n = k \cdot d $ , then $ \alpha ^ k, \alpha ^ { 2 k } , \alpha ^ { 3 k } , \ldots , \alpha ^ { dk } $ are all roots of $ x ^ d - 1 = 0 $
Every element with order dividing $ n $ , must be a power of $ \alpha $ , since an
element of order $ d $ is a $ d $ -th root of unity.
Every power of $ \alpha $ has order which divides $ n $ , and every field element
whose order divides $ n $ is a power of $ \alpha $ . This suggests that we partition
the powers of $ \alpha $ according to their orders:
$$ x ^ n - 1 = \prod _ { \stackrel { d, } { d|n } } \prod _ { \beta } ( x - \beta ) $$
where at each iteration, $ \beta $ is a field element of order $ d $ for each $ d $ .
The polynomial whose roots are the field elements of order $ d $ is called the
\emph { cyclotomic polynomial} , denoted by $ Q ^ { ( d ) } ( x ) $ .
\begin { thm} { 4.32}
$$ x ^ n - 1 = \prod _ { \stackrel { d, } { d|n } } Q ^ { ( d ) } ( x ) $$
\end { thm}
\subsubsection { From Ian Stewart's ``Galois Theory'' book}
Notes from Ian Stewart's book \cite { ianstewart} .
Examples:
Consider the case $ n = 12 $ , let $ \zeta = e ^ { \pi i / 6 } $ be a primitive $ 12 $ -th root of unity.
Classify its powers ($ \zeta ^ j $ ) according to their minimal power $ d $ such that
$ ( \zeta ^ j ) ^ d = 1 $ (ie. when they are primitive $ d $ -th roots of unity).
\begin { enumerate} []
\item $ d = 1 ,~~~ 1 $
\item $ d = 2 ,~~~ \zeta ^ 6 $
\item $ d = 3 ,~~~ \zeta ^ 4 , \zeta ^ 8 $
\item $ d = 4 ,~~~ \zeta ^ 3 , \zeta ^ 9 $
\item $ d = 6 ,~~~ \zeta ^ 2 , \zeta ^ { 10 } $
\item $ d = 12 ,~~~ \zeta , \zeta ^ 5 , \zeta ^ 7 , \zeta ^ { 11 } $
\end { enumerate}
Observe that we can factorize $ t ^ { 12 } - 1 $ by grouping the corresponding zeros:
\begin { align*}
t^ { 12} -1 = & (t-1) \times \\
& (t-\zeta ^ 6) \times \\
& (t-\zeta ^ 4) (t-\zeta ^ 8) \times \\
& (t-\zeta ^ 3) (t-\zeta ^ 9) \times \\
& (t-\zeta ^ 2) (t-\zeta ^ { 10} ) \times \\
& (t-\zeta ) (t-\zeta ^ 5)(t-\zeta ^ 7) (t-\zeta ^ { 11} )
\end { align*}
which simplifies to
$$ t ^ { 12 } - 1 = ( t - 1 ) ( t + 1 ) ( t ^ 2 + t + 1 ) ( t ^ 2 + 1 ) ( t ^ 2 - t + 1 ) F ( t ) $$
where $ F ( t ) = ( t - \zeta ) ( t - \zeta ^ 5 ) ( t - \zeta ^ 7 ) ( t - \zeta ^ { 11 } ) = t ^ 4 - t ^ 2 + 1 $ (this last step can be obtained either by multiplying $ ( t - \zeta ) ( t - \zeta ^ 5 ) ( t - \zeta ^ 7 ) ( t - \zeta ^ { 11 } ) $ together, or by dividing $ t ^ { 12 } - 1 $ by all the other factors).
Let $ \Phi _ d ( t ) $ be the factor corresponding to primitive $ d $ -th roots of unity, then we have proved that
$$ t ^ { 12 } - 1 = \Phi _ 1 \Phi _ 2 \Phi _ 3 \Phi _ 4 \Phi _ 6 \Phi _ { 12 } $$
\begin { defn} { 21.5}
The polynomial $ \Phi _ d ( t ) $ defined by
$$ \Phi _ n ( t ) = \prod _ { a \in \mathbb { Z } _ n, ( a,n ) = 1 } ( t - \zeta ^ a ) $$
is the $ n $ -th \emph { cyclotomic polynomial} over \mathbb { C} .
\end { defn}
\begin { cor} { 21.6}
$ \forall n \in \mathbb { N } $ , the polynomial $ \Phi _ n ( t ) $ lies in $ \mathbb { Z } [ t ] $ and is monic and irreducible.
\end { cor}
\begin { thm} { 21.9}
\begin { enumerate}
\item The Galois group $ \Gamma ( \mathbb { Q } ( \zeta ) : \mathbb { Q } ) $ consists of the
$ \mathbb { Q } $ -automorphisms $ \psi _ j $ defined by
$$ \psi _ j ( \zeta ) = \zeta ^ j $$
where $ 0 \leq j \leq n - 1 $ and $ j $ is prime to $ n $ .
\item $ \Gamma ( \mathbb { Q } ( \zeta ) : \mathbb { Q } ) \stackrel { iso } { \cong } \mathbb { Z } _ n ^ * $ , and is an abelian group.
\item its order is $ \phi ( n ) $
\item if $ n $ is prime, $ \mathbb { Z } _ n ^ * $ is cyclic
\end { thm}
\vspace { 1cm}
\subsubsection { Examples}
Examples of cyclotomic polynomials:
\begin { align*}
\Phi _ n(x) & = x^ { n-1} + x^ { n-2} + \ldots + x^ 2 + x + 1 = \sum _ { i=0} ^ { n-1} x^ i\\
@ -210,7 +330,7 @@ Examples:
\subsection { Lemma 1.42 from J.S.Milne's book}
\emph { TODO add reference to Milne's book}
Lemma from J.S.Milne's book \cite { milneFT} .
Useful for when dealing with $ x ^ p - 1 $ with $ p $ prime.
@ -249,7 +369,7 @@ Properties:
\begin { itemize}
\item are non-abelian (for $ n> 2 $ ), ie. $ rs \neq sr $
\item order $ 2 n $
\item generated by a rotation $ r $ and a reflex tion $ s $
\item generated by a rotation $ r $ and a reflec tion $ s $
\item $ r ^ n = s ^ 2 = id,~~~ ( rs ) ^ 2 = id $
\end { itemize}
Subgroups of $ \mathbb { D } _ n $ :
@ -281,7 +401,7 @@ For $n \geq 3, ~~\mathbb{D}_n \subseteq \mathbb{S}_n$ (subgroup of the Symmetry
\section { Exercises}
\subsection { Galois groups}
\subsection { Galois groups} \label { ex:galoisgroups}
\subsubsection [t6-7] { $ t ^ 6 - 7 \in \mathbb { Q } $ }
@ -364,7 +484,39 @@ $$\begin{aligned}
for $ 0 \leq k \leq 5 $ and $ j = \pm 1 $ .
\vspace { 0.5cm}
\emph { TODO diagram}
NOTE: WIP diagram.
\begin { tikzpicture} [node distance=2cm]
\def \radius { 2}
\draw (0,0) circle (\radius );
\foreach \k in { 0,...,5} {
% \node (a\k ) at ({ 360/6 * \k } :\radius ) { $ \alpha \zeta ^ { \k } $ } ;
\node (a\k ) at ({ 360/6 * \k } :\radius +0.5) { $ \alpha \zeta ^ { \k } $ } ;
\fill ({ 360/6 * \k } :\radius ) circle (2pt);
}
% real & im axis
\draw [->] (-2.5,0) -- (2.5,0) node[right] { } ;
\draw [->] (0,-2.5) -- (0,2.5) node[above] { } ;
% tau:
\draw [<->] (3,1) -- (3,-1) node[right] { $ \tau $ } ;
% sigma:
% \foreach \k [evaluate=\k as \next using int(mod(\k +1,6))] in { 0,...,5} {
% \coordinate (p\k ) at ({ 360/6 * \k } :\radius );
% \coordinate (p\next ) at ({ 360/6 * \next } :\radius );
%
% \draw [->, bend left=30] (p\k .center) -- node[above] { $ \sigma $ } (p\next .center);
% }
\foreach \k in { 0,...,5} {
\coordinate (p\k ) at ({ 360/6 * \k } :\radius );
}
\foreach \k [evaluate=\k as \next using int(mod(\k +1,6))] in { 0,...,5} {
\draw [->, bend left=30] (p\k ) -- node[above] { $ \sigma $ } (p\next );
}
\end { tikzpicture}
\vspace { 0.5cm}
Observe, that $ \Gamma $ is generated by the combination of $ \sigma $ and $ \tau $ ,
