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
In this paper we improve the character approach to the multiplier conjecture that we presented after 1992, and thus we have made considerable progress in the case of n = 3n1. We prove that in the case of n = 3n1 Secon...In this paper we improve the character approach to the multiplier conjecture that we presented after 1992, and thus we have made considerable progress in the case of n = 3n1. We prove that in the case of n = 3n1 Second multiplier theorem remains true if the assumption “n1 > λ” is replaced by “(n1, λ) = 1”. Consequentially we prove that if we let D be a (v, k, λ)-difference set in an abelian group G, and n = 3pr for some prime p, (p,v) = 1, then p is a numerical multiplier of D.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
基金This work was supported by the National Natural Science Foundation of China (Grant No. 19831070).
文摘In this paper we improve the character approach to the multiplier conjecture that we presented after 1992, and thus we have made considerable progress in the case of n = 3n1. We prove that in the case of n = 3n1 Second multiplier theorem remains true if the assumption “n1 > λ” is replaced by “(n1, λ) = 1”. Consequentially we prove that if we let D be a (v, k, λ)-difference set in an abelian group G, and n = 3pr for some prime p, (p,v) = 1, then p is a numerical multiplier of D.
