calculating the splitting of primes CalculatingTheSplittingOfPrimes 2003-08-21 20:45:30 2006-10-25 00:29:11 Topic % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newtheorem{thm}{Theorem} \newtheorem{prop}{Proposition} \newcommand{\ab}[1]{{#1}_{\mathrm{ab}}} \newcommand{\Ad}{\mathrm{Ad}} \newcommand{\ad}{\mathrm{ad}} \newcommand{\Aut}{\mathrm{Aut}\,} \newcommand{\Aff}[2]{\mathrm{Aff}_{#1} #2} \newcommand{\aff}[2]{\mathfrak{aff}_{#1} #2} \newcommand{\mcB}{\mathcal{B}} \newcommand{\p}{\mathfrak{p}} \renewcommand{\P}{\mathfrak{P}} \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\bfrac}[2]{\left[\frac{#1}{#2}\\right]} \newcommand{\bkh}{\backslash} \newcommand{\Cyc}[2]{\mathcal{C}^{#1}_{#2}} \newcommand{\Cbar}[2]{\overline{\C{#1}{#2}}} %\newcommand{\CD}{\R[\Delta]} \newcommand{\C}{\mathbb{C}} \newcommand{\CF}[2]{\ensuremath{\mathfrak{C}(#1,#2)}} \newcommand{\Cinf}{\EuScript{C}^{\infty}} \newcommand{\cmp}{cyclic mod $p$\xspace} \newcommand{\cp}{\mathrm{c.p.}} \newcommand{\CS}{\EuScript{CS}} \newcommand{\deck}{\EuScript{D}} \newcommand{\defl}[1]{\mathfrak{def}_{#1}} \newcommand{\Der}{\mathrm{Der}\,} \newcommand{\eH}{[X_H]-[Y_H]} \newcommand{\EL}{\mathcal{EL}} \newcommand{\End}{\mathrm{End}} \newcommand{\ES}[1]{\EuScript{#1}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Fix}{\mathrm{Fix}} \newcommand{\fr}[1]{\mathfrak{#1}} \newcommand{\Frat}{\mathrm{Frat}\,} \newcommand{\Gal}[1]{\Gamma(#1 |\Q)} \newcommand{\GL}[2]{\mathrm{GL}_{#1} #2} \newcommand{\gl}[2]{\mathfrak{gl}_{#1} #2} \newcommand{\GrR}[1]{a(#1 G)} \newcommand{\Gr}{\mathrm{Gr}\,} \newcommand{\mcH}{\mathcal{H}} \renewcommand{\H}{\mathbb{H}} \newcommand{\Hom}[2]{\mathrm{Hom}(#1,#2)} \newcommand{\id}{\mathrm{id}} \newcommand{\im}{\mathrm{im}} \newcommand{\ind}[2]{\mathrm{ind}^{#1}_{#2}} \newcommand{\indp}[2]{\mathfrak{ind}^{#1}_{#2}} \renewcommand{\inf}[1]{\mathfrak{inf}_{#1}} \newcommand{\inn}[1]{\langle #1\rangle} \renewcommand{\int}{\mathrm{int}} \newcommand{\Iso}{\mathrm{Iso}} \newcommand{\K}{\mathcal{K}} \renewcommand{\ker}{\mathrm{ker}\,} \renewcommand{\L}[1]{\mathfrak{L}(#1)} \newcommand{\lap}[1]{\Delta_{#1}} \newcommand{\lapM}{\Delta_M} \newcommand{\Lie}{\mathrm{Lie}} \newcommand{\lineq}{linearly equivalent\xspace} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mG}{m_G} \newcommand{\mK}{m_{\K}} \newcommand{\mindeg}[1]{\fr{md}(#1)} \newcommand{\N}{\mathbb{N}} \renewcommand{\O}{\mathcal{O}} \newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\Orb}{\mathrm{Orb}} \newcommand{\pad}{\hat{\Z}_p} \newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\pderw}[1]{\frac{\partial}{\partial #1}} \newcommand{\pdersec}[2]{\frac{\partial^2 #1}{\partial {#2}^2}} \newcommand{\perm}[1]{\pi_{#1}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\rad}{\mathrm{rad}\,} \newcommand{\res}[2]{\mathrm{res}^{#1}_{#2}} \newcommand{\resp}[2]{\mathfrak{res}^{#1}_{#2}} \newcommand{\RG}{\EuScript{R}_G} \newcommand{\rk}{\mathrm{rk}\,} \newcommand{\V}[1]{\mathbf{#1}} \newcommand{\vp}{\varphi} \newcommand{\Stab}{\mathrm{Stab}} \newcommand{\SL}[2]{\mathrm{SL}_{#1} #2} \renewcommand{\sl}[2]{\fr{sl}_{#1} #2} \newcommand{\SO}[2]{\mathrm{SO}_{#1} #2} \newcommand{\Sp}[2]{\mathrm{Sp}_{#1} #2} \renewcommand{\sp}[2]{\fr{sp}_{#1} #2} \newcommand{\SU}[1]{\mathrm{SU}( #1)} \newcommand{\su}[1]{\fr{su}_{#1}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\sym}{\mathrm{sym}} \newcommand{\Tg}{\mc{T}(\fr g)} \newcommand{\tom}{\tilde{\omega}} \newcommand{\ghtghp}{\fr g/\fr h\oplus(\fr g/\fr h^\perp)^*} \newcommand{\ghps}{(\fr g/\fr h^\perp)^*} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\tr}{\mathrm{tr}} %\renewcommand{\thechapter}{\Roman{chapter}} %\renewcommand{\thesection}{\thechapter.\arabic{section}} %\renewcommand{\thethm}{\thechapter.\arabic{thm}} \newcommand{\Ug}{\mc{U}(\fr g)} \newcommand{\Uh}{\mc{U}(\fr h)} \renewcommand{\V}[1]{\mathbf{#1}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Zp}{\Z/p} Let $K|L$ be an extension of number fields, with rings of integers $\O_K,\O_L$. Since this extension is \PMlinkname{separable}{SeparablePolynomial}, there exists $\alpha\in K$ with $L(\alpha)=K$ and by multiplying by a suitable integer, we may assume that $\alpha\in\O_K$ (we do {\em not} require that $\O_L[\alpha]=\O_K$. There is not, in general, an $\alpha\in\O_L$ with this property). Let $f\in \O_L[x]$ be the minimal polynomial of $\alpha$. Now, let $\p$ be a prime ideal of $L$ that does not divide $\Delta(f)\Delta(\O_K)^{-1}$, and let $\bar{f}\in \O_L/\p\O_L[x]$ be the reduction of $f$ mod $\p$, and let $\bar{f}=\bar{f}_1\cdots \bar{f}_n$ be its factorization into irreducible polynomials. If there are repeated factors, then $p$ splits in $K$ as the product $$\p=(\p,f_1(\alpha))\cdots(\p,f_n(\alpha)),$$ where $f_i$ is any polynomial in $\O_L[x]$ reducing to $\bar{f}_i$. Note that in this case $\p$ is unramified, since all $f_i$ are pairwise coprime mod $\p$ For example, let $L=\Q, K=\Q(\sqrt{d})$ where $d$ is a square-free integer. Then $f=x^2-d$. For any prime $\p$, $f$ is irreducible mod $\p$ if and only if it has no roots mod $\p$, i.e. $d$ is a quadratic non-residue mod $\p$. Using quadratic reciprocity, we can obtain a congruence condition mod $4p$ for which primes split and which do not. In general, this is possible for all fields with abelian Galois groups, using \PMlinkescapetext{class} field \PMlinkescapetext{theory}. Furthermore, let $K'$ be the splitting field of $L$. Then $G=\mathrm{Gal}(K'|L)$ acts on the roots of $f$, giving a map $G\to S_m$, where $m=\deg f$. Given a prime $\p$ of $\O_L$, the Artin symbol $[\P,K'|L]$ for any $\P$ lying over $\p$ is determined up to conjugacy by $\p$. Its \PMlinkescapetext{image} in $S_n$ is a product of disjoint cycles of length $m_1,\ldots,m_n$ where $m_i=\deg f_i$. This \PMlinkescapetext{information} is useful not just for prime splitting, but also for the calculation of Galois groups. Another useful fact is the Frobenius \PMlinkescapetext{density} theorem, which \PMlinkescapetext{states} that every element of $G$ is $[\P,K'|L]$ for infinitely many primes $\P$ of $\O_{K'}$. For example, let $f=x^3+x^2+2\in\Z[x]$. This is irreducible mod 3, and thus irreducible. Galois theory tells us that $G=\mathrm{Gal}(K'|L)$ is a subgroup of $S_3$, and so is isomorphic to $C_3$ or $S_3$, but it is not obvious which. But if we consider $p=7$, $f\equiv (x-2)(x^2+3x-1)\pmod 7$, and the quadratic factor is irreducible mod 7. Thus, $G\cong S_3$. Or let $f=x^4+ax^2+b$ for some integers $a,b$ and is irreducible. For a prime $p$, consider the factorization of $f$. Either it remains irreducible ($G$ contains a 4-cycle), splits as the product of irreducible quadratics ($G$ contains a cycle of the form $(12)(34)$) or $\bar{f}$ has a root. If $\beta$ is a root of $f$, then so is $-\beta$, and so assuming $p\neq 2$, there are at least two roots, and so a 3-cycle is impossible. Thus $G\cong C_4$ or $D_4$.