{"id":1557,"date":"2022-01-09T11:44:26","date_gmt":"2022-01-09T14:44:26","guid":{"rendered":"http:\/\/localhost\/?page_id=1557"},"modified":"2023-01-06T14:50:25","modified_gmt":"2023-01-06T17:50:25","slug":"o-anel-dos-polinomios","status":"publish","type":"page","link":"http:\/\/localhost\/index.php\/ensino\/fundamentos-de-algebra\/o-anel-dos-polinomios\/","title":{"rendered":"O anel dos polin\u00f4mios"},"content":{"rendered":"
\nSeja $R$ um anel e considere o conjunto dos polin\u00f4mios $R[x]$ sobre $R$. N\u00f3s introduzimos duas opera\u00e7\u00f5es $+$ e $\\cdot$ no conjunto $R[x]$. Sejam $f(x),g(x)\\in R[x]$ dados como
\n\\begin{align*}
\nf(x)&=\\alpha_nx^n+\\alpha_{n-1}x^{n-1}+\\cdots+\\alpha_1x+\\alpha_0\\\\
\ng(x)&=\\beta_nx^n+\\beta_{n-1}x^{n-1}+\\cdots+\\beta_1x+\\beta_0
\n\\end{align*}
\nonde $\\alpha_i,\\beta_i\\in R$. (Note que n\u00f3s n\u00e3o assumimos que $\\alpha_n\\neq 0$ e $\\beta_n\\neq 0$) ent\u00e3o os graus de $f(x)$ e $g(x)$ podem n\u00e3o coincidir.) Defina
\n\\begin{align*}
\nf(x)+g(x)&=(\\alpha_n+\\beta_n)x^n+(\\alpha_{n-1}+\\beta_{n-1})x^{n-1}+\\cdots+(\\alpha_1+\\beta_1)x+\\alpha_0+\\beta_0;\\\\
\nf(x)g(x)&=c_{2n}x^{2n}+c_{2n-1}x^{2n-1}+\\cdots +c_1x+c_0
\n\\end{align*}
\nonde
\n\\[
\nc_k=\\sum_{i,j\\in\\{0,\\ldots,n\\}\\\\ i+j=k} a_ib_j.
\n\\]<\/p>\n
\nConsidere $f(x)=x^3+\\overline 2x$ e $g(x)=\\overline 2x+\\overline 1$ em $\\Z_3[x]$. Ent\u00e3o
\n\\begin{align*}
\nf(x)+g(x) &= x^3+x+\\overline 1\\\\
\nf(x)g(x) & = (x^3+\\overline 2x)(\\overline 2x+\\overline 1)=\\overline 2x^4+x^3+x^2+\\overline 2x.
\n\\end{align*}<\/div>\n
\nSeja $R$ um anel e $f(x),g(x)\\in R[x]\\setminus\\{0\\}$.<\/p>\n