《Galois theory》 H.E. p. 59 (S44) * * * 证明的第六段 (最后一段)。 . When deg G/deg H = p, j in the above equation is 1, that is, const. G( X) = H(X, r)H(X,α·r)...H(X,α^(p-1)·r) ~~~ (1) ---- 当 deg G/deg H = p 时,上述方程中的 j = 1,即 ...
《Galois theory》 H.E. p. 59 (S44) * * * 12:29 证明的第四段。 Lemma . If U(X, r) = V(X, r)·W(X, r) where U, V, W are polynomials in two variables with coefficients in K (so that U(X, r), V(X, r), and V(X, r) are polynomials in one variable X with coefficients in K(r) = K') ...
《Galois theory》 H.E. p. 59 (S44) * * * 16:09 证明的第二段。 Since H(X) has coefficients in K' = K(r), where r is a pth root of k, and since every element of K' can be expressed as a polynomial in r with coefficients in K,... ---- 因为 H(x) 的系数在 K' = K(r) 中,其中 r ...
《Galois theory》 H.E. p. 59 (S44) * * * 16:00 第一段 Galois states in his Propositions II and III the main facts about the way in which the Galois group is reduced when the field of known quantities K is extended. ---- 伽罗瓦在他的命题 II 和 III 里陈述了关于当已知量的域 K 扩张时 ...
《Galois theory》 H.E. p. 56 (S42) * * * 12:52 2. Show that a subgroup H of a group G partitions a presentation of G into presentations of H and, in particular, that the number of elements in H divides the number of elements in G. ---- 群 G 的子群 H 分割 G 的表述为 H 的若干表述,并 ...
《Galois theory》 H.E. p. 56 (S42) * * * 14:53 2. Show that a subgroup H of a group G partitions a presentation of G into presentations of H and, in particular, that the number of elements in H divides the number of elements in G. ---- 群 G 的子群 H 分割 G 的表述为 H 的 ...
《Galois theory》 H.E. p. 56 (S42) * * * 15:23 Fifth Exercise Set 1. Show that a set of substitutions of n objects a, b, c, ... is a group if and only if it is closed under composition. ---- 置换的集合构成群当且仅当它在复合运算之下封闭。 . 证明:1) 置换的集合(记作 ...