French Film Theory and Criticism, Volume 1最新文献

Cinegraphie and the Search for Specificity 电影摄影学和对专一性的追求

French Film Theory and Criticism, Volume 1

Pub Date : 2020-12-08 DOI: 10.2307/j.ctv17db3jd.10

引用次数: 0

Selected Texts 选定的文本

French Film Theory and Criticism, Volume 1

Pub Date : 2020-12-08 DOI: 10.2307/j.ctv17db3jd.7

Paula S. Cohen

We give a short and self-contained proof of the Fundamental Theorem of Galois Theory (FTGT) for finite degree extensions. We derive the FTGT (for finite degree extensions) from two statements, denoted (a) and (b). These two statements, and the way they are proved here, go back at least to Emil Artin (precise references are given below). The argument is essentially taken from Chapter II of Emil Artin’s Notre Dame Lectures [A]. More precisely, statement (a) below is implicitly contained in the proof Theorem 10 page 31 of [A], in which the uniqueness up to isomorphism of the splitting field of a polynomial is verified. Artin’s proof shows in fact that, when the roots of the polynomial are distinct, the number of automorphisms of the splitting extension coincides with the degree of the extension. Statement (b) below is proved as Theorem 14 page 42 of [A]. The proof given here (using Artin’s argument) was written with Keith Conrad’s help. Theorem Let E/F be an extension of fields, let a1, . . . , an be distinct generators of E/F such that p := (X − a1) · · · (X − an) is in F [X]. Then • the group G of automorphisms of E/F is finite, • there is a bijective correspondence between the sub-extensions S/F of E/F and the subgroups H of G, and we have S ↔ H ⇐⇒ H = Aut(E/S) ⇐⇒ S = E =⇒ [E : S] = |H|, where E is the fixed subfield of H, where [E : S] is the degree (that is the dimension) of E over S, and where |H| is the order of H.

给出了有限次扩展下伽罗瓦理论基本定理的一个简短且完备的证明。我们从(a)和(b)两个表述中推导出FTGT(有限次扩展)。这两个表述以及它们在这里的证明方法，至少可以追溯到Emil Artin(精确的参考文献在下面给出)。这个论点基本上摘自埃米尔·阿廷的《巴黎圣母院讲座》第二章[A]。更确切地说，下面的表述(a)隐含地包含在[a]的证明定理10页31中，证明了多项式的分裂域的唯一性直到同构。Artin的证明实际上表明，当多项式的根不同时，分裂扩展的自同构数与扩展的程度一致。下面的表述(b)被证明为[A]第42页的定理14。这里给出的证明(使用Artin的论点)是在Keith Conrad的帮助下编写的。设E/F是域的扩展，设a1，…，可以是不同的E/F生成器，使得p:= (X−a1)···(X−an)在F [X]中。然后•同构的G组E / F是有限的,•之间有一个双射的对应关系复杂年代/ F (E / F和G的子组H, H和S↔⇐⇒H = Aut (E / S)⇐⇒S = E =⇒[E: S] = | | H, H E是固定的子域,在[E: S]的程度(即维度)E / S、H, | |是H。

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Index 指数

French Film Theory and Criticism, Volume 1

Pub Date : 2020-12-08 DOI: 10.2307/j.ctv17db3jd.14

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Back Matter 回到问题

French Film Theory and Criticism, Volume 1

Pub Date : 2020-12-08 DOI: 10.2307/j.ctv17db3jd.15

引用次数: 0

Photogenie and Company Photogenie公司

French Film Theory and Criticism, Volume 1

Pub Date : 2020-12-08 DOI: 10.2307/j.ctv17db3jd.8

引用次数: 1

Selected Texts 选定的文本

French Film Theory and Criticism, Volume 1

Pub Date : 2020-12-08 DOI: 10.17077/0084-9537.1314

M. Janco

引用次数: 0

The Great Debates 伟大的辩论

French Film Theory and Criticism, Volume 1

Pub Date : 2020-12-08 DOI: 10.2307/j.ctv17db3jd.12

The debate will entail the following format: Opening Arguments, Rebuttals (two rounds), Closing Arguments, and finally Class Discussion. Not only it is important to defend your position, superior debaters are ready to counter the arguments of the opposing side. And such arguments must be accompanied by tangible research (i.e. overheads, handouts, charts, graphs, pictures) to make for a more meaningful and exact debate.

辩论将包括以下形式:开场辩论，反驳(两轮)，结束辩论，最后是课堂讨论。不仅捍卫自己的立场很重要，优秀的辩手也准备好反驳对方的论点。这样的争论必须伴随着切实的研究(即管理费用、讲义、图表、图表、图片)，以使辩论更有意义和准确。

引用次数: 5

Acknowledgments 致谢

French Film Theory and Criticism, Volume 1

Pub Date : 2020-12-08 DOI: 10.2307/j.ctv17db3jd.5

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Selected Texts 选定的文本

French Film Theory and Criticism, Volume 1

Pub Date : 2020-12-08 DOI: 10.1515/9783110877281.339

P. Gaillard

We give a short and self-contained proof of the Fundamental Theorem of Galois Theory (FTGT) for finite degree extensions. We derive the FTGT (for finite degree extensions) from two statements, denoted (a) and (b). These two statements, and the way they are proved here, go back at least to Emil Artin (precise references are given below). The argument is essentially taken from Chapter II of Emil Artin’s Notre Dame Lectures [A]. More precisely, statement (a) below is implicitly contained in the proof Theorem 10 page 31 of [A], in which the uniqueness up to isomorphism of the splitting field of a polynomial is verified. Artin’s proof shows in fact that, when the roots of the polynomial are distinct, the number of automorphisms of the splitting extension coincides with the degree of the extension. Statement (b) below is proved as Theorem 14 page 42 of [A]. The proof given here (using Artin’s argument) was written with Keith Conrad’s help. Theorem Let E/F be an extension of fields, let a1, . . . , an be distinct generators of E/F such that p := (X − a1) · · · (X − an) is in F [X]. Then • the group G of automorphisms of E/F is finite, • there is a bijective correspondence between the sub-extensions S/F of E/F and the subgroups H of G, and we have S ↔ H ⇐⇒ H = Aut(E/S) ⇐⇒ S = E =⇒ [E : S] = |H|, where E is the fixed subfield ofH, where [E : S] is the degree (that is the dimension) of E over S, and where |H| is the order of H. Proof We claim: (a) If S/F is a sub-extension of E/F , then [E : S] = |Aut(E/S)|. (b) If H is a subgroup of G, then |H| = [E : E ]. Proof that (a) and (b) imply the theorem. Let S/F be a sub-extension of E/F and put H := Aut(E/S). Then we have trivially S ⊂ E , and (a) and (b) imply [E : S] = [E : E ]. 2 Conversely let H be a subgroup of G and set H := Aut(E/E). Then we have trivially H ⊂ H, and (a) and (b) imply |H| = |H|. Proof of (a). Let 1 ≤ i ≤ n. Put K := S[a1, . . . , ai−1] and d := [K[ai] : K]. It suffices to check that any F -embedding φ of K in E has exactly d extensions to an F -embedding Φ of K[ai] in E. Let q ∈ K[X] be the minimal polynomial of ai over K. It is enough to verify that φ(q) (the image under φ of q) has d distinct roots in E. But this is clear since q divides p, and thus φ(q) divides φ(p) = p. Proof of (b). In view of (a) it is enough to check |H| ≥ [E : E ]. Let k be an integer larger than |H|, and pick a b = (b1, . . . , bk) ∈ E. We must show that the bi are linearly dependent over E , or equivalently that b⊥ ∩ (E) is nonzero, where •⊥ denotes the vectors orthogonal to • in E with respect to the dot product on E. Any element of b⊥∩(EH)k is necessarily orthogonal to hb for any h ∈ H, so b⊥ ∩ (E) = (Hb)⊥ ∩ (E), where Hb is the H-orbit of b. We will show that (Hb)⊥ ∩ (E) is nonzero. Since the span of Hb in E has E-dimension at most |H| < k, (Hb)⊥ is nonzero. Choose a nonzero vector x in (Hb)⊥ such that xi = 0 for the largest number of i as possible among all nonzero vectors in (Hb)⊥. Some coordinate xj is no

给出了有限次扩展下伽罗瓦理论基本定理的一个简短且完备的证明。我们从(a)和(b)两个表述中推导出FTGT(有限次扩展)。这两个表述以及它们在这里的证明方法，至少可以追溯到Emil Artin(精确的参考文献在下面给出)。这个论点基本上摘自埃米尔·阿廷的《巴黎圣母院讲座》第二章[A]。更确切地说，下面的表述(a)隐含地包含在[a]的证明定理10页31中，证明了多项式的分裂域的唯一性直到同构。Artin的证明实际上表明，当多项式的根不同时，分裂扩展的自同构数与扩展的程度一致。下面的表述(b)被证明为[A]第42页的定理14。这里给出的证明(使用Artin的论点)是在Keith Conrad的帮助下编写的。设E/F是域的扩展，设a1，…，可以是不同的E/F生成器，使得p:= (X−a1)···(X−an)在F [X]中。然后•同构的G组E / F是有限的,•之间有一个双射的对应关系复杂年代/ F (E / F和G的子组H, H和S↔⇐⇒H = Aut (E / S)⇐⇒S = E =⇒[E: S] = | |,其中E是固定分区ofH, [E: S]在哪里的程度(即维度)E / S、H, | |是H .证明我们声称的顺序:(a)如果S / E / F, F是一个复杂问题然后Aut [E: S] = | | (E / S)。(b)若H是G的子群，则|H| = [E: E]。证明(a)和(b)包含定理。设S/F为E/F的子扩展，设H:= Aut(E/S)。那么我们就有S∧E， (a)和(b)暗示着[E: S] = [E: E]。反过来设H为G的子群，集H = Aut(E/E)。那么我们就有平凡的H∧H， (a)和(b)暗示|H| = |H|。(a)的证明。设1≤i≤n。令K:= S[a1，…]。， ai−1]和d:= [K[ai]: K]。它可以检查任何F嵌入φE正好有d K的扩展F嵌入ΦK (ai)在大肠让问∈K [X]是ai的最小多项式在K足以验证φ(q) (q)的图像在φd不等根自q将p E .但这是明确的,因此φ(q)将φ(p) = p。(b)的证据。针对H (a)它足以检查| |≥(E, E)。设k为大于|H|的整数，取b = (b1，…)我们必须证明bi在E上是线性相关的，或者等价地证明b⊥∩(E)是非零的，其中⊥表示E中关于E上点积的正交向量。对于任何h∈h, b⊥∩(EH)k的任何元素都必然正交于hb，因此b⊥∩(E) = (hb)⊥∩(E)，其中hb是b的h轨道。我们将证明(hb)⊥∩(E)是非零的。由于Hb在E中的张成空间最多有E维|H| < k，所以(Hb)⊥是非零的。在(Hb)⊥中选择一个非零向量x，使得在(Hb)⊥中的所有非零向量中，xi = 0的个数尽可能多。某些坐标xj在E中是非零的，因此通过缩放，我们可以假设对于某些j, xj = 1。由于E中的子空间(Hb)⊥在H的作用下是稳定的，对于H中的任何H，我们有hx∈(Hb)⊥，所以hx−x∈(Hb)⊥。由于xj = 1, hx−x的第j个坐标为0，所以hx−x = 0通过x的选择。由于h中的所有h都成立，x在(E)中。[A] Emil Artin，伽罗瓦理论，在圣母大学讲座，第二章:http://projecteuclid.org/euclid.ndml/1175197045。

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引用次数: 2

Before the Canon 在正典之前

French Film Theory and Criticism, Volume 1

Pub Date : 2020-12-08 DOI: 10.2307/j.ctv17db3jd.6

引用次数: 1