{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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}
{"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":null,"pages":null},"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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