diff --git a/galois-theory-notes.pdf b/galois-theory-notes.pdf index cab9619..1dcb87a 100644 Binary files a/galois-theory-notes.pdf and b/galois-theory-notes.pdf differ diff --git a/galois-theory-notes.tex b/galois-theory-notes.tex index aeb76ea..64fea4e 100644 --- a/galois-theory-notes.tex +++ b/galois-theory-notes.tex @@ -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, mostyly 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 $\alpha^k, \alpha^{2k}, \alpha^{3k}, \ldots, \alpha^{dk}$ 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 $2n$ - \item generated by a rotation $r$ and a reflextion $s$ + \item generated by a rotation $r$ and a reflection $s$ \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 $\tau$, diff --git a/notes_ntt.pdf b/notes_ntt.pdf index fa79dc6..f801442 100644 Binary files a/notes_ntt.pdf and b/notes_ntt.pdf differ diff --git a/notes_ntt.tex b/notes_ntt.tex index b8eec0b..c5c270d 100644 --- a/notes_ntt.tex +++ b/notes_ntt.tex @@ -34,8 +34,8 @@ The notes are not complete, don't include all the steps neither all the proofs. - An implementation of the NTT can be found at\\ - \href{https://github.com/arnaucube/fhe-study/blob/main/arithmetic/src/ntt.rs}{https://github.com/arnaucube/fhe-study/blob/main/arithmetic/src/ntt.rs}. + Update: an implementation of the NTT can be found at\\ + \href{https://github.com/arnaucube/fhe-study/blob/main/arith/src/ntt.rs}{https://github.com/arnaucube/fhe-study/blob/main/arith/src/ntt.rs}. \end{abstract} \tableofcontents