Pub Date : 2021-06-03DOI: 10.1017/9781108955911.019
{"title":"Normal and Separable Extensions, and Splitting Fields","authors":"","doi":"10.1017/9781108955911.019","DOIUrl":"https://doi.org/10.1017/9781108955911.019","url":null,"abstract":"","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":"65 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81090631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Definition 1.1. An extension L of a field k is said to be primary if the largest algebraic separable extension of k in L coincides with k. Proposition 1.2. Let X be a k-scheme. The following statements are equivalent. (a) For every extension K/k, X ⊗k K is irreducible,i.e., geometrically irreducible. (b) For every finite separable extension K/k, X ⊗k K is irreducible. (c) X is irreducible and if x is a generic point, k(x) is a primary extension of k. Proposition 1.3. Let Ω be an algebraically closed field of K and all extensions of K subextensions of ω. N a Galois extension of a field K, E any extension of K and L = N ∩ E. Then the fields E and N are linearly disjoint over L, i.e., E(N) ∼= E ⊗L N . Gal(E(N)/E) ∼= Gal(N/(E ∩ N))
{"title":"Galois Theory","authors":"A. Douady, R. Douady","doi":"10.1090/fim/021/01","DOIUrl":"https://doi.org/10.1090/fim/021/01","url":null,"abstract":"Definition 1.1. An extension L of a field k is said to be primary if the largest algebraic separable extension of k in L coincides with k. Proposition 1.2. Let X be a k-scheme. The following statements are equivalent. (a) For every extension K/k, X ⊗k K is irreducible,i.e., geometrically irreducible. (b) For every finite separable extension K/k, X ⊗k K is irreducible. (c) X is irreducible and if x is a generic point, k(x) is a primary extension of k. Proposition 1.3. Let Ω be an algebraically closed field of K and all extensions of K subextensions of ω. N a Galois extension of a field K, E any extension of K and L = N ∩ E. Then the fields E and N are linearly disjoint over L, i.e., E(N) ∼= E ⊗L N . Gal(E(N)/E) ∼= Gal(N/(E ∩ N))","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":"11 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88675825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-01DOI: 10.1142/S1005386721000195
Jia-Li Du, Yan-Quan Feng, Yu-Qin Liu
A graph [Formula: see text] is said to be symmetric if its automorphism group [Formula: see text] acts transitively on the arc set of [Formula: see text]. We show that if [Formula: see text] is a finite connected heptavalent symmetric graph with solvable stabilizer admitting a vertex-transitive non-abelian simple group [Formula: see text] of automorphisms, then either [Formula: see text] is normal in [Formula: see text], or [Formula: see text] contains a non-abelian simple normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] is explicitly given as one of 11 possible exceptional pairs of non-abelian simple groups. If [Formula: see text] is arc-transitive, then [Formula: see text] is always normal in [Formula: see text], and if [Formula: see text] is regular on the vertices of [Formula: see text], then the number of possible exceptional pairs [Formula: see text] is reduced to 5.
如果图[公式:见文]的自同构群[公式:见文]传递作用于[公式:见文]的弧集，则图[公式:见文]是对称的。我们证明了如果[公式:见文]是一个具有可解稳定子的有限连通七价对称图，它允许自同构的一个顶点传递的非阿贝尔单群[公式:见文]，那么[公式:见文]在[公式:见文]中是正规的，或者[公式:见文]包含一个非阿贝尔简单正规子群[公式:见文]使得[公式:见文]和[公式:见文]被明确地作为11个可能的非阿贝尔简单群例外对之一给出。如果[Formula: see text]是圆弧传递的，那么[Formula: see text]在[Formula: see text]中总是正常的，如果[Formula: see text]在[Formula: see text]的顶点上是规则的，那么可能的异常对[Formula: see text]的数量减少到5。
{"title":"Heptavalent Symmetric Graphs with Certain Conditions","authors":"Jia-Li Du, Yan-Quan Feng, Yu-Qin Liu","doi":"10.1142/S1005386721000195","DOIUrl":"https://doi.org/10.1142/S1005386721000195","url":null,"abstract":"A graph [Formula: see text] is said to be symmetric if its automorphism group [Formula: see text] acts transitively on the arc set of [Formula: see text]. We show that if [Formula: see text] is a finite connected heptavalent symmetric graph with solvable stabilizer admitting a vertex-transitive non-abelian simple group [Formula: see text] of automorphisms, then either [Formula: see text] is normal in [Formula: see text], or [Formula: see text] contains a non-abelian simple normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] is explicitly given as one of 11 possible exceptional pairs of non-abelian simple groups. If [Formula: see text] is arc-transitive, then [Formula: see text] is always normal in [Formula: see text], and if [Formula: see text] is regular on the vertices of [Formula: see text], then the number of possible exceptional pairs [Formula: see text] is reduced to 5.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":"46 1","pages":"243-252"},"PeriodicalIF":0.3,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85464327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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