{"id":409,"date":"2023-06-20T16:42:04","date_gmt":"2023-06-20T08:42:04","guid":{"rendered":"https:\/\/qwq.cafe\/?p=409"},"modified":"2023-06-20T16:42:04","modified_gmt":"2023-06-20T08:42:04","slug":"abstract-algebra-1-7-irreducibility","status":"publish","type":"post","link":"https:\/\/qwq.cafe\/?p=409","title":{"rendered":"Abstract Algebra \u2013 1.7 Irreducibility"},"content":{"rendered":"

Theorem 1.7.1<\/strong> If $f(x)=a_0+a_1x+\\cdots+a_nx^n\\in \\mathbb Z[x]\\subset \\mathbb Q[x]$, then every rational root of $f$ has the form $b\/c$, where $b\\mid a_0$ and $c\\mid a_n$. In particular, if $f$ is monic, then every rational root of $f$ is an integer.<\/p>\n

Theorem 1.7.2<\/strong> Let $f(x)=a_0+a_1x+\\cdots+a_{n-1}x^{n-1}+x^n\\in \\mathbb Z[x]$ be monic, and let $p$ be a prime. If $\\overline{f}(x)=[a_0]+[a_1]x+\\cdots+[a_{n-1}]x^{n-1}+x^n$ is irreducible in $\\mathbb F_p[x]$, then $f$ is irreducible in $\\mathbb Q[x]$.<\/p>\n

Definition 1.7.1<\/strong> If $n\\ge 1$ is a positive integer, then an $n$th root of unity<\/strong> in a field $k$ is an element $\\zeta\\in k$ with $\\zeta^k=1$. If $\\zeta$ is an $n$th root of unity and $n$ is the smallest positive integer for which $\\zeta^n=1$, we say that $\\zeta$ is a primitive $n$th root of unity<\/strong>.<\/p>\n

Definition 1.7.2<\/strong> If $d$ is a positive integer, then the $d$th cyclotomic polynomial<\/strong> is defined by
\n$$\\Phi_d(x)=\\prod(x-\\zeta),
\n$$
\nwhere $\\zeta$ ranges over all the primitive $d$th roots of unity.<\/p>\n

Proposition 1.7.1<\/strong> Let $n$ be a positive integer and regard $x^n-1\\in \\mathbb Z[x]$. Then<\/p>\n

    \n
  1. \n

    $$
    \nx^n-1=\\prod{d\\mid n}\\Phi<\/em>{d}(x),
    \n$$<\/p>\n

    where $d$ ranges over all the positive divisors $d$ of $n$.<\/p>\n<\/li>\n

  2. \n

    $\\Phi_n (x)$ is a monic polynomial in $\\mathbb Z[x]$ and $\\deg(\\Phi_n)=\\phi(n)$, the Euler $\\phi$-function.<\/p>\n<\/li>\n

  3. \n

    For every integer $MARKDOWN_HASH1d0fe3ad28ed27546808efeec5c626a9MARKDOWNHASH$, we have
    \n$$
    \nn=\\sum<\/em>{d\\mid n}\\phi(d).
    \n$$<\/p>\n<\/li>\n<\/ol>\n

    Proof.<\/p>\n

      \n
    1. \n

      For each divisor $d$ of $n$, collect all terms in the equation $x^n-1=\\prod(x-\\zeta)$ with $\\zeta$ a primitive $d$th root of unity.<\/p>\n<\/li>\n

    2. \n

      Let’s prove that $\\Phi_n(x)\\in \\mathbb Z[x]$ by induction on $n\\ge1$. The base step is true, for $\\Phi_1(x)=x-1\\in \\mathbb Z[x]$.<\/p>\n

      For the inductive step, let $f(x)=\\prod_{d\\mid n,d<n}\\Phi_d(x)$, so that
      \n$$
      \nx^n-1=f(x)\\Phi_n(x).
      \n$$
      \nBy induction, each $\\Phi_d(x)$ is a monic polynomial in $\\mathbb Z[x]$, and so $f$ is a monic polynomial in $\\mathbb Z[x]$.<\/p>\n

      Since $f$ and $x^n-1$ are monic, $\\Phi_n$ is a monic polynomial in $\\mathbb Z[x]$.<\/p>\n<\/li>\n

    3. \n

      Immediate from parts (1) and (2):
      \n$$
      \nn=\\deg(x^n-1)=\\deg(\\prod_d \\Phi_d)=\\sum_d \\deg(\\Phi_d)=\\sum_d \\phi(d)\\ \\square
      \n$$<\/p>\n<\/li>\n<\/ol>\n

      Corollary 1.7.1<\/strong> If $q$ is a positive integer and $d$ is a divisor of an integer $n$ with $d<n$, then $\\Phi_n(q)$ is a divisor of both $q^n-1$ and $(q^n-1)\/(q^d-1)$.<\/p>\n

      Theorem 1.7.3 (Eisenstein Criterion)<\/strong> Let $f(x)=a_0+a_1x+\\cdots+a_nx^n\\in \\mathbb Z[x]$. If there is a prime $p$ dividing $a_i$ for all $i<n$ but with $p\\nmid a_n$ and $p\\nmid a_0^2$, then $f$ is irreducible in $\\mathbb Q[x]$.<\/p>\n

      Theorem 1.7.4 (Gauss)<\/strong> For every prime $p$, the $p$th cyclotomic polynomial $\\Phi_p(x)$ is irreducible in $\\mathbb Q[x]$.<\/p>\n

      Proof.<\/p>\n

      Since $\\Phi_p(x)=(x^p-1)\/(x-1)$, we have
      \n$$\\Phi(x+1)=((x+1)^p-1)\/x=x^{p-1}+C_p^1x^{p-2}+C_p^2x^{p-3}+\\cdots+p.
      \n$$
      \nSince $p$ is prime, we have $p\\mid C_p^i$ for all $i\\ (0<i<p)$; hence, Eisenstein Criterion applies, and $\\Phi(x+1)$ is irreducible in $\\mathbb Q[x]$. $\\square$<\/p>\n","protected":false},"excerpt":{"rendered":"

      Theorem 1.7.1 If $f(x)=a_0+a_1x+\\cdots+a_nx^n\\in \\mathb […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12,6],"tags":[11,8],"class_list":["post-409","post","type-post","status-publish","format-standard","hentry","category-abstract-algebra","category-6","tag-11","tag-8"],"_links":{"self":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/409","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=409"}],"version-history":[{"count":1,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/409\/revisions"}],"predecessor-version":[{"id":410,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/409\/revisions\/410"}],"wp:attachment":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=409"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=409"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=409"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}