{"id":395,"date":"2023-06-20T16:32:15","date_gmt":"2023-06-20T08:32:15","guid":{"rendered":"https:\/\/qwq.cafe\/?p=395"},"modified":"2023-06-20T16:32:15","modified_gmt":"2023-06-20T08:32:15","slug":"abstract-algebra-1-1-polynomials","status":"publish","type":"post","link":"https:\/\/qwq.cafe\/?p=395","title":{"rendered":"Abstract Algebra \u2013 1.1 Polynomials"},"content":{"rendered":"<p><strong>Definition 1.1.1<\/strong> If $R$ is a commutative ring, then a <strong>formal power series<\/strong> over $R$ is a sequence of elements $s_i\\in R$ for all $i\\ge 0$, called the <strong>coefficients<\/strong> of $\\sigma$:<br \/>\n$$\\sigma = (s_0,s_1,s_2,\\dots,s_i,\\dots).<br \/>\n$$<br \/>\nWe can consider $\\sigma$ as a function from $\\mathbb N$ to $R$. Denote $R[[x]]$ as the set of all formal power series over $R$.<\/p>\n<p>Let $\\sigma=(s_0,s_1,s_2,\\dots,s_i,\\dots)$ and $\\tau=(t_0,t_1,t_2,\\dots,t_i,\\dots)$ be formal power series.<\/p>\n<p>We define two binary operations, addition and multiplication, on $R[[x]]$.<br \/>\n$$\\sigma +\\tau := (s_0+t_0,s_1+t_1,\\dots,s_i+t_i,\\dots);\\\\<br \/>\n\\sigma\\cdot \\tau := (s_0t_0,s_0t_1+s_1t_0,\\dots,\\sum_{j=0}^is_jt_{i-j},\\dots).<br \/>\n$$<\/p>\n<p><strong>Definition 1.1.2<\/strong> A <strong>polynomial<\/strong> over a commutative ring $R$ is a formal power series $\\sigma$ over $R$ for which there exists an integer $n\\ge 0$ with $\\sigma(i)=0,\\forall i&gt;n$. <\/p>\n<p>A polynomial has only finitely many nonzero coefficients. The <strong>zero polynomial<\/strong>, denoted by $\\sigma = 0$, is the sequence $\\sigma = (0,0,0,\\dots)$.<\/p>\n<p><strong>Definition 1.1.3<\/strong> If $\\sigma=(s_0,s_1,\\dots,s_n,0,0,\\dots)$ is a nonzero polynomial, then there is $n\\ge 0$ with $s_n\\ne 0$ and $s_i=0,\\forall i&gt;n$. We call $s_n$ the <strong>leading coefficient<\/strong> of $\\sigma$, we call $n$ the <strong>degree<\/strong> of $\\sigma$, and we denote the degree by<br \/>\n$$n = \\deg(\\sigma)<br \/>\n$$<br \/>\nIf the leading coefficient $s_n=1$, the $\\sigma$ is called <strong>monic<\/strong>.<\/p>\n<p>Specifically, the zero polynomial $0$ does not have a degree because it has no nonzero coefficients.<\/p>\n<p>Denote $R[x]$ as the set of all polynomials over $R$. It&#8217;s trivial that $R[x]\\subset R[[x]]$.<\/p>\n<p><strong>Proposition 1.1.1<\/strong> If $R$ is a commutative ring, then $R[[x]]$ is a commutative ring that contains $R[x]$ and $R&#039;$ as subrings, where $R&#039;=\\{(r,0,0,\\dots);r\\in R\\}$.<\/p>\n<p><strong>Lemma 1.1.1<\/strong> Let $R$ be a commutative ring and let $\\sigma,\\tau\\in R$ be nonzero polynomials.<\/p>\n<ul>\n<li>Either $\\sigma\\tau=0$ or $\\deg(\\sigma\\tau)\\le \\deg(\\sigma)+\\deg(\\tau)$.<\/li>\n<li>If $R$ is a domain, then $\\sigma\\tau\\ne 0$ and $\\deg(\\sigma\\tau)=\\deg(\\sigma)+\\deg(\\tau)$.<\/li>\n<li>If $R$ is a domain, $\\sigma,\\tau\\ne 0$ and $\\tau\\mid\\sigma$ in $R[x]$, then $\\deg(\\tau)\\le \\deg(\\sigma)$.<\/li>\n<li>If $R$ is a domain, then $R[x]$ is a domain.<\/li>\n<\/ul>\n<p><strong>Definition 1.1.4<\/strong> Let $R$ be a commutative ring. The <strong>indeterminate<\/strong> $x\\in R[x]$ is<br \/>\n$$x=(0,1,0,0,\\dots).<br \/>\n$$<br \/>\nAfter defining indeterminate, we can denote a formal power series as $s_0+s_1x+s_2x^2+\\cdots+s_ix^i+\\cdots$.<\/p>\n<p>Now we can describe the usual role of $x$ in $f(x)$ as a variable. Each polynomial $f(x)=s_0+s_1x+s_2x^2+\\cdots+s_nx^n\\in R[x]$ defines a <strong>polynomial function<\/strong><br \/>\n$$f^\\flat:R\\to R<br \/>\n$$<br \/>\nby evaluation: If $a\\in R$, $f^\\flat(a)=s_0+s_1a+s_2a^2+\\cdots+s_na^n\\in R$. It should be realized that polynomial and polynomial function are distinct objects. <\/p>\n<p>Sometimes, we write $f^\\flat$ as $f$.<\/p>\n<p><strong>Definition 1.1.5<\/strong> Let $K$ be a field. The fraction field $\\operatorname{Frac}(K[x])$ of $K[x]$, denoted by $K(x)$, is called the <strong>field of rational functions<\/strong> over $K$.<\/p>\n<p><strong>Proposition 1.1.2<\/strong> If $K$ is a field, then the elements of $K(x)$ have the form $f(x)\/g(x)$, where $f(x),g(x)\\in K[x]$ and $g(x)\\ne 0$.<\/p>\n<p>We usually call $R[x]$ the ring of all polynomials over $R$ in one variable, but also there exist polynomials over $R$ in more than one variables.<\/p>\n<p><strong>Definition 1.1.6<\/strong> Let $R$ be a commutative ring, $R[x_1,x_2,\\dots,x_n]$ is the ring of <strong>polynomials over $R$ in $n$ variables<\/strong>. When $n\\ge 2$,<br \/>\n$$R[x_1,x_2,\\dots,x_n]:=(R[x_1,x_2,\\dots,x_{n-1}])[x_n].<br \/>\n$$<br \/>\nMoreover, when $K$ is a field, we can describe $\\operatorname{Frac}(K[x_1,x_2,\\dots,x_n])$ as all <strong>rational functions in $n$ variables<\/strong><br \/>\n$$K(x_1,x_2,\\dots,x_n).<br \/>\n$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Definition 1.1.1 If $R$ is a commutative ring, then a f [&hellip;]<\/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-395","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\/395","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=395"}],"version-history":[{"count":1,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/395\/revisions"}],"predecessor-version":[{"id":396,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/395\/revisions\/396"}],"wp:attachment":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=395"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=395"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=395"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}