It is proved in this paper that Abel’s and Galois’s proofs that the quintic equations have no radical solutions are invalid. Due to Abel’s and Galois’s work about two hundred years ago, it was generally accepted t...It is proved in this paper that Abel’s and Galois’s proofs that the quintic equations have no radical solutions are invalid. Due to Abel’s and Galois’s work about two hundred years ago, it was generally accepted that general quintic equations had no radical solutions. However, Tang Jianer <i><span style="font-family:Verdana;font-size:12px;">et</span></i><i><span style="font-size:12px;font-family:Verdana;"> al</span><span style="font-size:12px;font-family:Verdana;">.</span></i><span style="font-size:10pt;font-family:;" "=""><span style="font-family:Verdana;font-size:12px;"> recently prove that there are radical solutions for some quintic equations with special forms. The theories of Abel and Galois cannot explain these results. On the other hand, Gauss </span><i><span style="font-family:Verdana;font-size:12px;">et</span></i></span><i><span style="font-size:12px;font-family:Verdana;"> al</span><span style="font-size:12px;font-family:Verdana;">.</span></i><span style="font-size:10pt;font-family:;" "=""><span style="font-family:Verdana;font-size:12px;"> proved the fundamental theorem of algebra. The theorem declared that there were </span><i><span style="font-family:Verdana;font-size:12px;">n</span></i><span style="font-family:Verdana;font-size:12px;"> solutions for the </span><i><span style="font-family:Verdana;font-size:12px;">n</span></i><span style="font-family:Verdana;font-size:12px;"> degree equations, including the radical and non-radical solutions. The theories of Abel and Galois contradicted with the fundamental theorem of algebra. Due to the reasons above, the proofs of Abel and Galois should be re-examined and re-evaluated. The author carefully analyzed the Abel’s original paper and found some serious mistakes. In order to prove that the general solution of algebraic equation</span></span><span style="font-size:10pt;font-family:;" "=""> </span><span style="font-size:12px;font-family:Verdana;">he proposed was effective for the cubic equation, Abel took the known solutions of cubic equation as a pr展开更多
文摘It is proved in this paper that Abel’s and Galois’s proofs that the quintic equations have no radical solutions are invalid. Due to Abel’s and Galois’s work about two hundred years ago, it was generally accepted that general quintic equations had no radical solutions. However, Tang Jianer <i><span style="font-family:Verdana;font-size:12px;">et</span></i><i><span style="font-size:12px;font-family:Verdana;"> al</span><span style="font-size:12px;font-family:Verdana;">.</span></i><span style="font-size:10pt;font-family:;" "=""><span style="font-family:Verdana;font-size:12px;"> recently prove that there are radical solutions for some quintic equations with special forms. The theories of Abel and Galois cannot explain these results. On the other hand, Gauss </span><i><span style="font-family:Verdana;font-size:12px;">et</span></i></span><i><span style="font-size:12px;font-family:Verdana;"> al</span><span style="font-size:12px;font-family:Verdana;">.</span></i><span style="font-size:10pt;font-family:;" "=""><span style="font-family:Verdana;font-size:12px;"> proved the fundamental theorem of algebra. The theorem declared that there were </span><i><span style="font-family:Verdana;font-size:12px;">n</span></i><span style="font-family:Verdana;font-size:12px;"> solutions for the </span><i><span style="font-family:Verdana;font-size:12px;">n</span></i><span style="font-family:Verdana;font-size:12px;"> degree equations, including the radical and non-radical solutions. The theories of Abel and Galois contradicted with the fundamental theorem of algebra. Due to the reasons above, the proofs of Abel and Galois should be re-examined and re-evaluated. The author carefully analyzed the Abel’s original paper and found some serious mistakes. In order to prove that the general solution of algebraic equation</span></span><span style="font-size:10pt;font-family:;" "=""> </span><span style="font-size:12px;font-family:Verdana;">he proposed was effective for the cubic equation, Abel took the known solutions of cubic equation as a pr
