Regular extension

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In field theory, a branch of algebra, a field extension is said to be regular if k is algebraically closed in L (i.e., where is the set of elements in L algebraic over k) and L is separable over k, or equivalently, is an integral domain when is the algebraic closure of (that is, to say, are linearly disjoint over k).[1][2]

Properties[edit]

  • Regularity is transitive: if F/E and E/K are regular then so is F/K.[3]
  • If F/K is regular then so is E/K for any E between F and K.[3]
  • The extension L/k is regular if and only if every subfield of L finitely generated over k is regular over k.[2]
  • Any extension of an algebraically closed field is regular.[3][4]
  • An extension is regular if and only if it is separable and primary.[5]
  • A purely transcendental extension of a field is regular.

Self-regular extension[edit]

There is also a similar notion: a field extension is said to be self-regular if is an integral domain. A self-regular extension is relatively algebraically closed in k.[6] However, a self-regular extension is not necessarily regular.[citation needed]

References[edit]

  1. ^ Fried & Jarden (2008) p.38
  2. ^ a b Cohn (2003) p.425
  3. ^ a b c Fried & Jarden (2008) p.39
  4. ^ Cohn (2003) p.426
  5. ^ Fried & Jarden (2008) p.44
  6. ^ Cohn (2003) p.427
  • Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 11 (3rd revised ed.). Springer-Verlag. pp. 38–41. ISBN 978-3-540-77269-9. Zbl 1145.12001.
  • M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese) [1]
  • Cohn, P. M. (2003). Basic Algebra. Groups, Rings, and Fields. Springer-Verlag. ISBN 1-85233-587-4. Zbl 1003.00001.
  • A. Weil, Foundations of algebraic geometry.