可解な5次方程式の計算例

$$x^5-5x-12=0$$
の場合
\begin{align} D&=2^{12}\c5^6\\ -E,-\E&=2^5\c5^3,\ 2^7\c5^3\\ F&=5+\sqrt5\\ \\ \a_0&=\frac{\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=\frac{125}4(3+\sqrt5)\l(25+5\sqrt5+6\sqrt{5(5+\sqrt5)}\r)\\ \g_2&=\frac{125}4(3-\sqrt5)\l(25-5\sqrt5+6\sqrt{5(5-\sqrt5)}\r)\\ \g_3&=\frac{125}4(3-\sqrt5)\l(25-5\sqrt5-6\sqrt{5(5-\sqrt5)}\r)<0\\ \g_4&=\frac{125}4(3+\sqrt5)\l(25+5\sqrt5-6\sqrt{5(5+\sqrt5)}\r)\\ \\ \sqrt[5]{\g_1\g_4}&=5\sqrt5\\ \sqrt[5]{\g_2\g_3}&=-5\sqrt5\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=25\l(-\sqrt5+\sqrt{5+\sqrt5}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=25\l(-\sqrt5-\sqrt{5+\sqrt5}\r)\\ \end{align}

$$x^5-x^4-4x^3-8x^2+3x-23=0$$
の場合
\begin{align} D&=2^{18}\c11^4\\ -E,-\E&=2^9\c11^2,\ 2^9\c11^2\\ F&=5+\sqrt5\\ \\ \a_0&=\frac{1+\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=22(1+3\sqrt5)\l(18+16\sqrt5+5\sqrt{5(5+\sqrt5)}\r)\\ \g_2&=22(1-3\sqrt5)\l(18-16\sqrt5-5\sqrt{5(5-\sqrt5)}\r)\\ \g_3&=22(1-3\sqrt5)\l(18-16\sqrt5+5\sqrt{5(5-\sqrt5)}\r)<0\\ \g_4&=22(1+3\sqrt5)\l(18+16\sqrt5-5\sqrt{5(5+\sqrt5)}\r)\\ \\ \sqrt[5]{\g_1\g_4}&=11(1+\sqrt5)\\ \sqrt[5]{\g_2\g_3}&=11(1-\sqrt5)\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=22\l(3-\sqrt5+\sqrt{5(5+\sqrt5)}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=22\l(3-\sqrt5-\sqrt{5(5+\sqrt5)}\r) \end{align}

$$x^5-5x^3+10x-4=0$$
の場合
\begin{align} D&=2^6\c5^6\\ -E,-\E&=2^2\c5^3,\ 2^4\c5^3\\ F&=2(5+\sqrt5)\\ \\ \a_0&=\frac{\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=\frac{125}4\l(-20\sqrt5+(3+\sqrt5)\sqrt{10(5+\sqrt5)}\r)<0\\ \g_2&=\frac{125}4\l(\phantom{{}-}20\sqrt5+(3-\sqrt5)\sqrt{10(5-\sqrt5)}\r)\\ \g_3&=\frac{125}4\l(\phantom{{}-}20\sqrt5-(3-\sqrt5)\sqrt{10(5-\sqrt5)}\r)\\ \g_4&=\frac{125}4\l(-20\sqrt5-(3+\sqrt5)\sqrt{10(5+\sqrt5)}\r)<0\\ \\ \sqrt[5]{\g_1\g_4}&=5\c\frac{5-\sqrt5}2\\ \sqrt[5]{\g_2\g_3}&=5\c\frac{5+\sqrt5}2\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=\frac{25}2\l(2\sqrt5+\sqrt{2(5+\sqrt5)}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=\frac{25}2\l(2\sqrt5-\sqrt{2(5+\sqrt5)}\r) \end{align}

$$x^5-2x^4-6x^3+10x^2+17x-12=0$$
の場合
\begin{align} D&=2^{10}\c19^4\\ -E,-\E&=2^4\c19^2,\ 2^6\c19^2\\ F&=2(5+\sqrt5)\\ \\ \a_0&=\frac{2+\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=\frac{19}4(1+2\sqrt5)\l(202+46\sqrt5+15(3+\sqrt5)\sqrt{2(5+\sqrt5)}\r)\\ \g_2&=\frac{19}4(1-2\sqrt5)\l(202-46\sqrt5+15(3-\sqrt5)\sqrt{2(5-\sqrt5)}\r)<0\\ \g_3&=\frac{19}4(1-2\sqrt5)\l(202-46\sqrt5-15(3-\sqrt5)\sqrt{2(5-\sqrt5)}\r)<0\\ \g_4&=\frac{19}4(1+2\sqrt5)\l(202+46\sqrt5+15(3+\sqrt5)\sqrt{2(5+\sqrt5)}\r)\\ \\ \sqrt[5]{\g_1\g_4}&=19\\ \sqrt[5]{\g_2\g_3}&=19\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=\frac{19}2\l(-1-5\sqrt5+\sqrt{10(5+\sqrt5)}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=\frac{19}2\l(-1-5\sqrt5-\sqrt{10(5+\sqrt5)}\r) \end{align}

$$x^5+10x^3-15x^2+10x-12=0$$
の場合
\begin{align} D&=2^2\c3^2\c5^8\c11^2\\ -E,-\E&=2^2\c3\c5^4,\ 3\c5^4\c11^2\\ F&=6(5+\sqrt5)\\ \\ \a_0&=\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}\\ \g_1&=3+\sqrt5+\frac{1+\sqrt5}4\sqrt{6(5+\sqrt5)}\\ \g_2&=3-\sqrt5+\frac{1-\sqrt5}4\sqrt{6(5-\sqrt5)}<0\\ \g_3&=3-\sqrt5-\frac{1-\sqrt5}4\sqrt{6(5-\sqrt5)}\\ \g_4&=3+\sqrt5-\frac{1+\sqrt5}4\sqrt{6(5+\sqrt5)}<0\\ \\ \sqrt[5]{\g_1\g_4}&=-1\\ \sqrt[5]{\g_2\g_3}&=-1\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=\frac14\l(3+\sqrt5-\sqrt{6(5+\sqrt5)}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=\frac14\l(3+\sqrt5+\sqrt{6(5+\sqrt5)}\r)\\ \end{align}

$$x^5-5x^2-3=0$$
の場合
\begin{align} D&=3^4\c5^6\\ -E,-\E&=3\c5^3,\ 3^3\c5^3\\ F&=6(5+\sqrt5)\\ \\ \a_0&=\frac{\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=\frac{125}8\l(10(5+\sqrt5)+(3+\sqrt5)\sqrt{30(5+\sqrt5)}\r)\\ \g_2&=\frac{125}8\l(10(5-\sqrt5)+(3-\sqrt5)\sqrt{30(5-\sqrt5)}\r)\\ \g_3&=\frac{125}8\l(10(5-\sqrt5)-(3-\sqrt5)\sqrt{30(5-\sqrt5)}\r)\\ \g_4&=\frac{125}8\l(10(5+\sqrt5)-(3+\sqrt5)\sqrt{30(5+\sqrt5)}\r)<0\\ \\ \sqrt[5]{\g_1\g_4}&=-5\sqrt5\\ \sqrt[5]{\g_2\g_3}&=5\sqrt5\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=25\l(5+\sqrt5-\sqrt{6(5-\sqrt5)}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=25\l(5+\sqrt5+\sqrt{6(5-\sqrt5)}\r) \end{align}

$$x^5-2x^4+6x^3-9x^2+4x+1=0$$
の場合
\begin{align} D&=7^2\c11^4\\ -E,-\E&=7\c11^2,\ 7\c11^2\\ F&=14(5+\sqrt5)\\ \\ \a_0&=\frac{2-\sqrt[5]{\g_1}-\sqrt[5]{\g_2}-\sqrt[5]{\g_3}-\sqrt[5]{\g_4}}5\\ \g_1&=\frac{11}4(4+\sqrt5)\l(53+43\sqrt5+15\sqrt{14(5+\sqrt5)}\r)\\ \g_2&=\frac{11}4(4-\sqrt5)\l(53-43\sqrt5+15\sqrt{14(5-\sqrt5)}\r)\\ \g_3&=\frac{11}4(4-\sqrt5)\l(53-43\sqrt5-15\sqrt{14(5-\sqrt5)}\r)<0\\ \g_4&=\frac{11}4(4+\sqrt5)\l(53+43\sqrt5-15\sqrt{14(5+\sqrt5)}\r)<0\\ \\ \sqrt[5]{\g_1\g_4}&=-11\\ \sqrt[5]{\g_2\g_3}&=-11\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=\frac{11}4\l(-7-5\sqrt5+\sqrt{70(5+\sqrt5)}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=\frac{11}4\l(-7-5\sqrt5-\sqrt{70(5+\sqrt5)}\r) \end{align}

$$x^5-5x^2+10x-4=0$$
の場合
\begin{align} D&=2^2\c5^6\c7^2\\ -E,-\E&=5^3\c7,\ 2^2\c5^3\c7\\ F&=14(5+\sqrt5)\\ \\ \a_0&=-\frac{\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=\frac{125}4(5+3\sqrt5)\l(10+\sqrt{14(5+\sqrt5)}\r)\\ \g_2&=\frac{125}4(5-3\sqrt5)\l(10-\sqrt{14(5-\sqrt5)}\r)<0\\ \g_3&=\frac{125}4(5-3\sqrt5)\l(10+\sqrt{14(5-\sqrt5)}\r)<0\\ \g_4&=\frac{125}4(5+3\sqrt5)\l(10-\sqrt{14(5+\sqrt5)}\r)<0\\ \\ \sqrt[5]{\g_1\g_4}&=-5\sqrt5\\ \sqrt[5]{\g_2\g_3}&=5\sqrt5\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=\frac{25}4\l(-5-3\sqrt5+\sqrt{14(5+\sqrt5)}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=\frac{25}4\l(-5-3\sqrt5-\sqrt{14(5+\sqrt5)}\r)\\ \end{align}

$$x^5+5x^3-5x^2+5x+3=0$$
の場合
\begin{align} D&=3^6\c5^6\c11^2\\ -E,-\E&=5^3\c11,\ 3^2\c5^3\c11\\ F&=\sqrt{22(5+\sqrt5)}\\ \\ \a_0&=\frac{\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=\frac{125}8\l(\phantom{{}-}60\sqrt5+(15-\sqrt5)\sqrt{22(5+\sqrt5)}\r)\\ \g_2&=\frac{125}8\l(-60\sqrt5-(15+\sqrt5)\sqrt{22(5-\sqrt5)}\r)<0\\ \g_3&=\frac{125}8\l(-60\sqrt5+(15+\sqrt5)\sqrt{22(5-\sqrt5)}\r)\\ \g_4&=\frac{125}8\l(\phantom{{}-}60\sqrt5+(15-\sqrt5)\sqrt{22(5+\sqrt5)}\r)<0\\ \\ \sqrt[5]{\g_1\g_4}&=-5\c\frac{5+\sqrt5}2\\ \sqrt[5]{\g_2\g_3}&=-5\c\frac{5-\sqrt5}2\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=\frac{25}4\l(5-3\sqrt5-\sqrt{22(5-\sqrt5)}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=\frac{25}4\l(5-3\sqrt5+\sqrt{22(5-\sqrt5)}\r) \end{align}

$$x^5-5x^3-10x^2+50x-80=0$$
の場合
\begin{align} D&=2^8\c5^6\c13^4\\ -E,-\E&=2^2\c5^3\c13^3,\ 2^5\c5^3\c13\\ F&=26(5+\sqrt5)\\ \\ \a_0&=\frac{\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=\frac{125}4\l(10(25+17\sqrt5)+(15+\sqrt5)\sqrt{26(5-\sqrt5)}\r)\\ \g_2&=\frac{125}4\l(10(25-17\sqrt5)+(15-\sqrt5)\sqrt{26(5+\sqrt5)}\r)\\ \g_3&=\frac{125}4\l(10(25-17\sqrt5)-(15-\sqrt5)\sqrt{26(5+\sqrt5)}\r)<0\\ \g_4&=\frac{125}4\l(10(25+17\sqrt5)-(15+\sqrt5)\sqrt{26(5-\sqrt5)}\r)\\ \\ \sqrt[5]{\g_1\g_4}&=5\c\frac{5+7\sqrt5}2\\ \sqrt[5]{\g_2\g_3}&=5\c\frac{5-7\sqrt5}2\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=\frac{25}4\l(10(1-\sqrt5)+(1+\sqrt5)\sqrt{26(5+\sqrt5)}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=\frac{25}4\l(10(1-\sqrt5)-(1+\sqrt5)\sqrt{26(5+\sqrt5)}\r) \end{align}

$$x^5+5bx^3+5b^2x-a=0$$
の場合
\begin{align} D&=5^5(a^2+4b^5)^2\\ -E,-\E&=(a^2+4b^5)\l(\frac{5\pm\sqrt5}2\r)^5\\ F&=a^2+4b^5\\ \\ \a_0&=\sqrt[5]{\g_1}+\sqrt[5]{\g_4}\\ \g_1&=\frac{a+\sqrt{a^2+4b^5}}2\\ \g_4&=\frac{a-\sqrt{a^2+4b^5}}2\\ \\ \sqrt[5]{\g_1\g_4}&=-b \end{align}

$$x^5-x^4+4x^3+4x^2-x+13=0$$
の場合
\begin{align} D&=2^8\c3^4\c5^3\\ -E,-\E&=2^4\c3^2\c5\sqrt5\c\bigg(\frac{\sqrt5\pm1}2\bigg)^7\\ F&=1\\ \\ \a_0 &=\frac15\l(1-\sqrt[5]{2^3\c3}+\sqrt[5]{2^6\c3^2}-\sqrt[5]{2^9\c3^3}-\sqrt[5]{2^2\c3^4}\r)\\ &=-\frac{(\sqrt[5]{24})^4+4}{4(\sqrt[5]{24}+1)} \end{align}

$$x^5-2x^4+6x^3+2x^2+4x+1=0$$
の場合
\begin{align} D&=5^3\c11^4\\ -E,-\E&=5\sqrt5\c11^2\c\bigg(\frac{\sqrt5\pm1}2\bigg)^5\\ F&=5\\ \\ \a_0&=\frac{2+\sqrt[5]{\g_1}-\sqrt[5]{\g_2}-\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=11^2\c\frac{9+5\sqrt5}2\\ \g_2&=11\c\bigg(\frac{9-5\sqrt5}2\bigg)^3<0\\ \g_3&=11\c\bigg(\frac{9+5\sqrt5}2\bigg)^3\\ \g_4&=11^2\c\frac{9-5\sqrt5}2<0\\ \\ \sqrt[5]{\g_1\g_4}&=-11\\ \sqrt[5]{\g_2\g_3}&=-11 \end{align}

$$x^5-2x^4+x^3+2x^2-x-4=0$$
の場合
\begin{align} D&=2^6\c3^4\c5^3\\ -E,-\E &=2^3\c3^2\c\l(\frac{5\pm\sqrt5}2\r)^3\\ F&=10\\ \\ \a_0&=\frac{2-\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}-\sqrt[5]{\g_4}}5\\ \g_1&=12(14+5\sqrt{10})\\ \g_2&=9(14+5\sqrt{10})^2/2\\ \g_3&=9(14-5\sqrt{10})^2/2\\ \g_4&=12(14-5\sqrt{10})<0\\ \\ \sqrt[5]{\g_1\g_4}&=-6\\ \sqrt[5]{\g_2\g_3}&=9 \end{align}

$$x^5-x^4-2x^3+6x^2-3x-13=0$$
の場合
\begin{align} D&=2^8\c5^3\c13^2\\ -E,-\E&=2^4\c5\sqrt5\c13\c\bigg(\frac{\sqrt5\pm1}2\bigg)^7\\ F&=5\c13\\ \\ \a_0&=\frac{1-\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}-\sqrt[5]{\g_4}}5\\ \g_1&=2(37+5\sqrt{65})\\ \g_2&=\g_1^2\\ \g_3&=\g_4^2\\ \g_4&=2(37-5\sqrt{65})<0\\ \\ \sqrt[5]{\g_1\g_4}&=-4\\ \sqrt[5]{\g_2\g_3}&=16 \end{align}

$$x^5-2x^4+4x^3-8x^2+11x-10=0$$
の場合
\begin{align} D&=2^{10}\c5^3\c7^2\\ -E,-\E &=2^5\c5\c7\c\frac{5\pm\sqrt5}2\\ F&=2\c5\c7\\ \\ \a_0&=\frac{2-\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}-\sqrt[5]{\g_4}}5\\ \g_1&=49(7+5\sqrt{70})\\ \g_2&=(7+5\sqrt{70})^2/7\\ \g_3&=(7-5\sqrt{70})^2/7\\ \g_4&=49(7-5\sqrt{70})<0\\ \\ \sqrt[5]{\g_1\g_4}&=-21\\ \sqrt[5]{\g_2\g_3}&=9 \end{align}

$$x^5-2x^4+2x^3+8x^2-8x+4=0$$
の場合
\begin{align} D&=2^8\c3^2\c5^3\c7^2\\ -E,-\E &=2^4\c3\c5\c7\c\frac{5\pm\sqrt5}2\\ F&=3\c5\c7\\ \\ \a_0&=\frac{2-\sqrt[5]{\g_1}-\sqrt[5]{\g_2}-\sqrt[5]{\g_3}-\sqrt[5]{\g_4}}5\\ \g_1&=18(51+5\sqrt{105})\\ \g_2&=4(51+5\sqrt{105})^2/3\\ \g_3&=4(51-5\sqrt{105})^2/3\\ \g_4&=18(51-5\sqrt{105})<0\\ \\ \sqrt[5]{\g_1\g_4}&=-6\\ \sqrt[5]{\g_2\g_3}&=4 \end{align}

$$x^5-2x^4+6x^3-8x^2+9x-14=0$$
の場合
\begin{align} D&=2^{17}\c5^3\\ -\E &=2^8\c5\c(10+3\sqrt{10})\\ F&=5(2+\sqrt2)\\ \\ \a_0&=\frac{2+\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=857+50\sqrt2+5(89+17\sqrt2)\sqrt{10(2+\sqrt2)}\\ \g_2&=857-50\sqrt2-5(89-17\sqrt2)\sqrt{10(2-\sqrt2)}\\ \g_3&=857-50\sqrt2+5(89-17\sqrt2)\sqrt{10(2-\sqrt2)}\\ \g_4&=857+50\sqrt2-5(89+17\sqrt2)\sqrt{10(2+\sqrt2)}<0\\ \\ \sqrt[5]{\g_1\g_4}&=-11-10\sqrt2\\ \sqrt[5]{\g_2\g_3}&=-11+10\sqrt2\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=13+40\sqrt2-5(4-\sqrt2)\sqrt{5(2+\sqrt2)}\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=13+40\sqrt2+5(4-\sqrt2)\sqrt{5(2+\sqrt2)}\\ \end{align}

$$x^5-2x^4-2x^3+8x^2-x-10=0$$
の場合
\begin{align} D&=2^{11}\c3^4\\ -\E&=2^5\sqrt2\c3^2\c(\sqrt2+1)^5\\ F&=10+3\sqrt{10}\\ \\ \a_0&=\frac{2+\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=3\l(1219+395\sqrt{10}+5(59+16\sqrt{10})\sqrt{10+3\sqrt{10}}\r)\\ \g_2&=3\l(1219-395\sqrt{10}+5(59-16\sqrt{10})\sqrt{10-3\sqrt{10}}\r)\\ \g_3&=3\l(1219-395\sqrt{10}-5(59-16\sqrt{10})\sqrt{10-3\sqrt{10}}\r)<0\\ \g_4&=3\l(1219+395\sqrt{10}-5(59+16\sqrt{10})\sqrt{10+3\sqrt{10}}\r)\\ \\ \sqrt[5]{\g_1\g_4}&=3(3+\sqrt{10})\\ \sqrt[5]{\g_2\g_3}&=3(3-\sqrt{10})\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=3\l(-9-4\sqrt{10}+5\sqrt{10+3\sqrt{10}}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=3\l(-9-4\sqrt{10}-5\sqrt{10+3\sqrt{10}}\r) \end{align}

$$x^5-x^4+4x^3+4x^2+2x-2=0$$
の場合
\begin{align} D&=2^{11}\c3^4\c5^2\\ -\E&=2^5\c3^2\c5\c(2+\sqrt2)\\ F&=10+3\sqrt{10}\\ \\ \a_0&=\frac{1+\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=3\l(1267+410\sqrt{10}+25(11+4\sqrt{10})\sqrt{10+3\sqrt{10}}\r)\\ \g_2&=3\l(1267-410\sqrt{10}-25(11-4\sqrt{10})\sqrt{10-3\sqrt{10}}\r)<0\\ \g_3&=3\l(1267-410\sqrt{10}+25(11-4\sqrt{10})\sqrt{10-3\sqrt{10}}\r)<0\\ \g_4&=3\l(1267+410\sqrt{10}-25(11+4\sqrt{10})\sqrt{10+3\sqrt{10}}\r)<0\\ \\ \sqrt[5]{\g_1\g_4}&=3(-3-\sqrt{10})\\ \sqrt[5]{\g_2\g_3}&=3(-3+\sqrt{10})\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=3\l(-13-3\sqrt{10}+5\sqrt{10+3\sqrt{10}}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=3\l(-13-3\sqrt{10}-5\sqrt{10+3\sqrt{10}}\r) \end{align}

$$x^5-2x^4-17x^3+8x^2+38x-48=0$$
の場合
\begin{align} D&=2^{14}\c5^3\c13^3\\ -\E&=2\c5\c13\c(\sqrt{65}-1)^2(65+8\sqrt{65})\\ F&=5(13+3\sqrt{13})\\ \\ \a_0&=\frac{2+\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=\frac12\l(1887+475\sqrt{13}+10(16+3\sqrt{13})\sqrt{5(13+3\sqrt{13})}\r)\\ \g_2&=\frac12\l(1887-475\sqrt{13}+10(16-3\sqrt{13})\sqrt{5(13-3\sqrt{13})}\r)\\ \g_3&=\frac12\l(1887-475\sqrt{13}-10(16-3\sqrt{13})\sqrt{5(13-3\sqrt{13})}\r)\\ \g_4&=\frac12\l(1887+475\sqrt{13}-10(16+3\sqrt{13})\sqrt{5(13+3\sqrt{13})}\r)\\ \\ \sqrt[5]{\g_1\g_4}&=\frac{93+25\sqrt{13}}2\\ \sqrt[5]{\g_2\g_3}&=\frac{93-25\sqrt{13}}2\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=\frac12\l(171+25\sqrt{13}+5(5+\sqrt{13})\sqrt{5(13-3\sqrt{13})}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=\frac12\l(171+25\sqrt{13}-5(5+\sqrt{13})\sqrt{5(13-3\sqrt{13})}\r) \end{align}

$$x^5-2x^4+3x^3-37x^2-27x-133=0$$
の場合
\begin{align} D&=3^8\c5^3\c7^2\c13^3\\ -\E&=3^4\c5\c7\c13\c(65+8\sqrt{65})\\ F&=70(13+3\sqrt{13})\\ \\ \a_0&=\frac{2+\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=\frac18\l(4(89889+14950\sqrt{13})+45(29+79\sqrt{13})\sqrt{70(13+3\sqrt{13})}\r)\\ \g_2&=\frac18\l(4(89889-14950\sqrt{13})+45(29-79\sqrt{13})\sqrt{70(13-3\sqrt{13})}\r)\\ \g_3&=\frac18\l(4(89889-14950\sqrt{13})-45(29-79\sqrt{13})\sqrt{70(13-3\sqrt{13})}\r)\\ \g_4&=\frac18\l(4(89889+14950\sqrt{13})-45(29+79\sqrt{13})\sqrt{70(13+3\sqrt{13})}\r)<0\\ \\ \sqrt[5]{\g_1\g_4}&=\frac{-7-15\sqrt{13}}2\\ \sqrt[5]{\g_2\g_3}&=\frac{-7+15\sqrt{13}}2\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2}&=\frac14\l(867+295\sqrt{13}-45\sqrt{70(13+3\sqrt{13})}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3}&=\frac14\l(867+295\sqrt{13}+45\sqrt{70(13+3\sqrt{13})}\r) \end{align}

$$x^5-2x^4+2x^3+3x^2-3x+4=0$$
の場合
\begin{align} D&=5^3\c17^3\\ -\E&=5\c17\c\frac{85+\sqrt{85}}2\\ F&=17+4\sqrt{17}\\ \\ \a_0&=\frac{2-\sqrt[5]{\g_1}-\sqrt[5]{\g_2}-\sqrt[5]{\g_3}-\sqrt[5]{\g_4}}5\\ \g_1&=\frac12\l(5261+1250\sqrt{17}+25(39+8\sqrt{17})\sqrt{17+4\sqrt{17}}\r)\\ \g_2&=\frac12\l(5261-1250\sqrt{17}-25(39-8\sqrt{17})\sqrt{17-4\sqrt{17}}\r)<0\\ \g_3&=\frac12\l(5261-1250\sqrt{17}+25(39-8\sqrt{17})\sqrt{17-4\sqrt{17}}\r)\\ \g_4&=\frac12\l(5261+1250\sqrt{17}-25(39+8\sqrt{17})\sqrt{17+4\sqrt{17}}\r)<0\\ \\ \sqrt[5]{\g_1\g_4}&=-1\\ \sqrt[5]{\g_2\g_3}&=-1\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2} &=\frac14\l(103+25\sqrt{17}-5(3+\sqrt{17})\sqrt{17+4\sqrt{17}}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3} &=\frac14\l(103+25\sqrt{17}+5(3+\sqrt{17})\sqrt{17+4\sqrt{17}}\r) \end{align}

$$x^5-2x^4-2x^3-2x^2+18x-18=0$$
の場合
\begin{align} D&=2^4\c3^2\c5^3\c17^3\\ -\E&=-2\c5\c17\c(85+7\sqrt{85})\\ F&=5(17+4\sqrt{17})\\ \\ \a_0&=\frac{2+\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=\frac12\l(4889+1325\sqrt{17}+45(11-\sqrt{17})\sqrt{5(17+4\sqrt{17})}\r)\\ \g_2&=\frac12\l(4889-1325\sqrt{17}-45(11+\sqrt{17})\sqrt{5(17-4\sqrt{17})}\r)<0\\ \g_3&=\frac12\l(4889-1325\sqrt{17}+45(11+\sqrt{17})\sqrt{5(17-4\sqrt{17})}\r)\\ \g_4&=\frac12\l(4889+1325\sqrt{17}-45(11-\sqrt{17})\sqrt{5(17+4\sqrt{17})}\r)\\ \\ \sqrt[5]{\g_1\g_4}&=9+5\sqrt{17}\\ \sqrt[5]{\g_2\g_3}&=9-5\sqrt{17}\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2} &=\frac12\l(71-15\sqrt{17}+5(1-\sqrt{17})\sqrt{5(17+4\sqrt{17})}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3} &=\frac12\l(71-15\sqrt{17}-5(1-\sqrt{17})\sqrt{5(17+4\sqrt{17})}\r) \end{align}

$$x^5-2x^4-6x^3+12x^2+36x+20=0$$
の場合
\begin{align} D&=2^8\c5^3\c17^3\\ -E&=2^3\c5\c17\c(85+9\sqrt{85})\\ F&=5(17+4\sqrt{17})\\ \\ \a_0&=\frac{2+\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=-2543+2025\sqrt{17}-5\c\frac{1-39\sqrt{17}}2\sqrt{10(17+\sqrt{17})}\\ \g_2&=-2543-2025\sqrt{17}+5\c\frac{1+39\sqrt{17}}2\sqrt{10(17-\sqrt{17})}<0\\ \g_3&=-2543-2025\sqrt{17}-5\c\frac{1+39\sqrt{17}}2\sqrt{10(17-\sqrt{17})}<0\\ \g_4&=-2543+2025\sqrt{17}+5\c\frac{1-39\sqrt{17}}2\sqrt{10(17+\sqrt{17})}<0\\ \\ \sqrt[5]{\g_1\g_4}&=19-5\sqrt{17}\\ \sqrt[5]{\g_2\g_3}&=19+5\sqrt{17}\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2} &=2\l(-11-15\sqrt{17}+5\sqrt{10(17+\sqrt{17})}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3} &=2\l(-11-15\sqrt{17}-5\sqrt{10(17+\sqrt{17})}\r) \end{align}

$$x^5-x^4+2x^3-4x^2+x-1=0$$
の場合
\begin{align} D&=2^4\c13^3\\ -\E&=26(13+3\sqrt{13})\\ F&=65+8\sqrt{65}\\ \\ \a_0&=\frac{1+\sqrt[5]{\g_1}+\sqrt[5]{\g_2}+\sqrt[5]{\g_3}+\sqrt[5]{\g_4}}5\\ \g_1&=\frac12\l(1477+125\sqrt{65}+5(25+2\sqrt{65})\sqrt{2(65+\sqrt{65})}\r)\\ \g_2&=\frac12\l(1477-125\sqrt{65}-5(25-2\sqrt{65})\sqrt{2(65-\sqrt{65})}\r)<0\\ \g_3&=\frac12\l(1477-125\sqrt{65}+5(25-2\sqrt{65})\sqrt{2(65-\sqrt{65})}\r)\\ \g_4&=\frac12\l(1477+125\sqrt{65}-5(25+2\sqrt{65})\sqrt{2(65+\sqrt{65})}\r)<0\\ \\ \sqrt[5]{\g_1\g_4}&=-4\\ \sqrt[5]{\g_2\g_3}&=-4\\ (\sqrt[5]{\g_4})^2\sqrt[5]{\g_2} &=\frac14\l(74+10\sqrt{65}-(5+\sqrt{65})\sqrt{2(65+\sqrt{65})}\r)\\ (\sqrt[5]{\g_1})^2\sqrt[5]{\g_3} &=\frac14\l(74+10\sqrt{65}+(5+\sqrt{65})\sqrt{2(65+\sqrt{65})}\r) \end{align}

$$x^5-10x^3-5x^2+10x-1=0$$
の場合
\begin{align} D&=5^8\c7^2\\ E,\E&=5^4,\ 5^4\c7^2\\ -F&=10+2\sqrt5\\ \\ \a &=\d_1+\d_2+\d_3+\d_4\\ &=2\cos\frac{(10j+1)\pi}{25}+2\cos\frac{(20j+7)\pi}{25}\qquad(j=0,1,2,3,4)\\ \d_1^5&=\z_{10}\ =\frac{1+\sqrt5+\sqrt{-10+2\sqrt5}}4\\ \d_2^5&=\z_{10}^{-3}=\frac{1-\sqrt5-\sqrt{-10-2\sqrt5}}4\\ \d_3^5&=\z_{10}^3\ =\frac{1-\sqrt5+\sqrt{-10-2\sqrt5}}4\\ \d_4^5&=\z_{10}^{-1}=\frac{1+\sqrt5-\sqrt{-10+2\sqrt5}}4\\ \\ \d_1\d_4&=1\\ \d_2\d_3&=1\\ \d_2&=\z_{10}\c\d_1^2\\ \d_3&=\z_{10}^{-2}\c\d_1^3 \end{align}

$$x^5-x^4-4x^3+3x^2+3x-1=0$$
の場合
\begin{align} D&=11^4\\ E,\E&=11^2,\ 11^2\\ -F&=10+2\sqrt5\\ \\ \a&=\frac{1+\d_1+\d_2+\d_3+\d_4}5\\ &=2\cos\frac{(2j+1)\pi}{11}\qquad(j=0,1,2,3,4)\\ \d_1^5&=\frac{11}4(3-2\sqrt5)\l(-47-23\sqrt5-5\sqrt{-10+2\sqrt5}\r)\\ \d_2^5&=\frac{11}4(3+2\sqrt5)\l(-47+23\sqrt5+5\sqrt{-10-2\sqrt5}\r)\\ \d_3^5&=\frac{11}4(3+2\sqrt5)\l(-47+23\sqrt5-5\sqrt{-10-2\sqrt5}\r)\\ \d_4^5&=\frac{11}4(3-2\sqrt5)\l(-47-23\sqrt5+5\sqrt{-10+2\sqrt5}\r)\\ \\ \d_1\d_4&=11\\ \d_2\d_3&=11\\ \d_4^2\d_2&=\frac{11}4\l(-1-5\sqrt5+\sqrt{5(-10+2\sqrt5)}\r)\\ \d_1^2\d_3&=\frac{11}4\l(-1-5\sqrt5-\sqrt{5(-10+2\sqrt5)}\r) \end{align}

$$x^5-25x^3-10x^2+50x-20=0$$
の場合
\begin{align} d&=2^{10}\c5^6\c7^4\\ E,\E&=2^3\c5^3\c7,\ 2^7\c5^3\c7^3\\ -F&=7(5+\sqrt5)\\ \\ \a&=\frac{\d_1+\d_2+\d_3+\d_4}5\\ \d_1^5&=\frac{125}2\l(125+95\sqrt5+(65-9\sqrt5)\sqrt{7(-5+\sqrt5)}\r)\\ \d_2^5&=\frac{125}2\l(125-95\sqrt5+(65+9\sqrt5)\sqrt{7(-5-\sqrt5)}\r)\\ \d_3^5&=\frac{125}2\l(125-95\sqrt5-(65+9\sqrt5)\sqrt{7(-5-\sqrt5)}\r)\\ \d_4^5&=\frac{125}2\l(125+95\sqrt5-(65-9\sqrt5)\sqrt{7(-5+\sqrt5)}\r)\\ \\ \d_1\d_4&=5\c\frac{25-\sqrt5}2\\ \d_2\d_3&=5\c\frac{25+\sqrt5}2\\ \d_4^2\d_2&=\frac{25}2\l(5+5\sqrt5-(1+3\sqrt5)\sqrt{7(-5+\sqrt5)}\r)\\ \d_1^2\d_3&=\frac{25}2\l(5+5\sqrt5+(1+3\sqrt5)\sqrt{7(-5+\sqrt5)}\r) \end{align}

$$x^5-20x^3-5x^2+90x+49=0$$
の場合
\begin{align} D&=3^6\c6^6\c37^2\\ E,\E&=3^2\c5^3\c37,\ 3^4\c5^3\c37\\ F&=74(5+\sqrt5)\\ \\ \a&=\frac{\d_1+\d_2+\d_3+\d_4}5\\ \d_1^5&=\frac{125}4\l(-75+165\sqrt5+(20-7\sqrt5)\sqrt{74(-5+\sqrt5)}\r)\\ \d_2^5&=\frac{125}4\l(-75-165\sqrt5+(20+7\sqrt5)\sqrt{74(-5-\sqrt5)}\r)\\ \d_3^5&=\frac{125}4\l(-75-165\sqrt5-(20+7\sqrt5)\sqrt{74(-5-\sqrt5)}\r)\\ \d_4^5&=\frac{125}4\l(-75+165\sqrt5-(20-7\sqrt5)\sqrt{74(-5+\sqrt5)}\r)\\ \\ \d_1\d_4&=50-5\sqrt5\\ \d_2\d_3&=50+5\sqrt5\\ \d_4^2\d_2&=\frac{25}4\l(5-23\sqrt5+\sqrt{74(-5+\sqrt5)}\r)\\ \d_1^2\d_3&=\frac{25}4\l(5-23\sqrt5-\sqrt{74(-5+\sqrt5)}\r) \end{align}

$$x^5-20x^3+60x-32=0$$
の場合
\begin{align} D&=2^{21}\c5^6\\ E,\E&=2^{10}\sqrt2\c5^3\c(\sqrt2\pm1)^3\\ -F&=10+\sqrt{10}\\ \\ \a&=\frac{\d_1+\d_2+\d_3+\d_4}5\\ \d_1^5&=500\l(-50+15\sqrt{10}+(10-3\sqrt{10})\sqrt{-10+\sqrt{10}}\r)\\ \d_2^5&=500\l(-50-15\sqrt{10}-(10+3\sqrt{10})\sqrt{-10-\sqrt{10}}\r)\\ \d_3^5&=500\l(-50-15\sqrt{10}+(10+3\sqrt{10})\sqrt{-10-\sqrt{10}}\r)\\ \d_4^5&=500\l(-50+15\sqrt{10}-(10-3\sqrt{10})\sqrt{-10+\sqrt{10}}\r)\\ \\ \d_1\d_4&=50-10\sqrt{10}\\ \d_2\d_3&=50+10\sqrt{10}\\ \d_4^2\d_2&=50\l(\sqrt{10}-\sqrt{2(-10+3\sqrt{10})}\r)\\ \d_1^2\d_3&=50\l(\sqrt{10}+\sqrt{2(-10+3\sqrt{10})}\r) \end{align}

$$x^5-x^4-11x^3+4x^2+29x+13=0$$
の場合
\begin{align} D&=3^4\c5^3\c13^3\\ E,\E&=3^2\c5\c13\c(65\pm8\sqrt{65})\\ F&=2(13+3\sqrt{13})\\ \\ \a&=\frac{1+\d_1+\d_2+\d_3+\d_4}5\\ \d_1^5&=\frac38(47-13\sqrt{13})\l(-309+89\sqrt{13}+25\sqrt{2(-13+3\sqrt{13})}\r)\\ \d_2^5&=\frac38(47+13\sqrt{13})\l(-309-89\sqrt{13}+25\sqrt{2(-13-3\sqrt{13})}\r)\\ \d_3^5&=\frac38(47+13\sqrt{13})\l(-309-89\sqrt{13}-25\sqrt{2(-13-3\sqrt{13})}\r)\\ \d_4^5&=\frac38(47-13\sqrt{13})\l(-309+89\sqrt{13}-25\sqrt{2(-13+3\sqrt{13})}\r)\\ \\ \d_1\d_4&=\frac{57-15\sqrt{13}}2\\ \d_2\d_3&=\frac{57+15\sqrt{13}}2\\ \d_4^2\d_2&=\frac38(1-\sqrt{13})\l(7-3\sqrt{13}+5\sqrt{2(-13+3\sqrt{13})}\r)\\ \d_1^2\d_3&=\frac38(1-\sqrt{13})\l(7-3\sqrt{13}-5\sqrt{2(-13+3\sqrt{13})}\r) \end{align}