《Galois theory》 H.E. p. 52 * * * 17:30 (略去一段) . Proposition 1 . Let Ψ(U, V, W, ...) be a polynomial in n variables with coefficients in K. ---- 设立系数在 K 中的 n 元多项式 Ψ(U, V, W, ...) . . Let Ψ t' be the element of K(a, b, c,...) obtained by setting U = ...
《Galois cohomology》 少一点思考就多一点效率 ~ * * * 9:50 Let K' 1 /K 1 and K' 2 /K 2 be two Galois extensions, with Galois groups G 1 and G 2 . ---- 取两个 Galois 扩展,及对应的 Galois 群. . &nbs ...
《Galois theory》 Galois 群是根的排列构成的群. * * * 17:30 Using this lemma, Galois proves (A) as follows. . Since f( φ a (X)) is a polynomial in X with coefficients in K, and since it has the root t in common with the irreducible polynomial G(X), by Lemma I it ...
《Galois cohomology》 适当地放慢速度 ~ * * * 16:20 Let A be a functor verifying the above axioms. ---- 令 A 为函子且符合三条公理. . 评论:此处直接从函子出发. . If K'/K is a Galois extension, the H^q(Gal(K'/K), A(K')) are defined. ---- 若 K'/K ...
《Galois theory》 * * * 21:20 Proof. Let the Euclidean algorithm be used to write a common divisor d(X) of g(X) and h(X) in the form d(X) = A(X)g(X) + B(X)h(X) where A(X) and B(X) are polynomials with coefficients in K. ---- 写出 g 和 h 的公共因式 d 的形式... ---- d = Ag + Bh ( ...
《Galois cohomology》 在“待定”学习法中学习者不必理解每个知识点,正如在代数运算中允许存在未知元那样. * * * 15:20 Remarks. 1) If k s denotes a separable closure of k, the group A(k s ) is well-defined, and it is a Gal(k s /k)-group. ---- 若 ks 是 k ...
《Galois theory》 * * * 18:55 In his proof of (A), and in several later proofs, Galois makes use of a simple and very basic lemma: ---- (A) 的证明及其它证明中,Galois 使用了一个简单且非常基本的引理: . Lemma 1 . If g(X) and h(X) are polynomials with coefficients in a given fi ...
《Galois cohomology》 * * * 16:50 ... and this functor verifies the following axioms: ---- 这一函子符合如下公理: . 评论:这里所说的函子就是 A(K). . (1) A(K) = lim A(K i ), for K i running over the set of sub-extensions of K of finite type over k. - ...
《Galois theory》 终于给出 Galois 群... * * * 2:30 Finally, let G(X) be an irreducible (over K) factor of F(X) of which t is a root. (See S38) ---- 令 G(X) 是 F(X) 的不可约因式,t 是它的根. ---- “不可约”是关于 K 而言. . 评论:前文说 F(X) 是 n! 次多项式且有 n! 个不同的 ...