On differential Galois groups of strongly normal extensions
نویسندگان
چکیده
منابع مشابه
The Differential Galois Theory of Strongly Normal Extensions
Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished. This may be due to its reliance on Kolchin’s elegant, but not widely adopted, axiomatization of the theory of algebraic groups. This paper attempts to revive the theory using a differential scheme in place of those axioms. We also avoid using a universal differential field, instead relying on a c...
متن کاملGalois Groups of Maximal ̂ -extensions
Let p be an odd prime and F a field of characteristic different from p containing a primitive p\h root of unity. Assume that the Galois group G of the maximal p-extension of F has a finite normal series with abelian factor groups. Then the commutator subgroup of G is abelian. Moreover, G has a normal abelian subgroup with pro-cyclic factor group. If, in addition, F contains a primitive p2th roo...
متن کاملGalois Groups of Radical Extensions
Theorem 1.1 (Kummer theory). Let m ∈ Z>0, and suppose that the subgroup μm(K) = {ζ ∈ K∗ : ζ = 1} of K∗ has order m. Write K∗1/m for the subgroup {x ∈ K̄∗ : x ∈ K∗} of K̄∗. Then K(K∗1/m) is the maximal abelian extension of exponent dividing m of K inside K̄, and there is an isomorphism Gal(K(K∗1/m)/K) ∼ −→ Hom(K∗, μm(K)) that sends σ to the map sending α to σ(β)/β, where β ∈ K∗1/m satisfies β = α.
متن کاملOn Galois Groups of Abelian Extensions over Maximal Cyclotomic Fields
Let k0 be a finite algebraic number field in a fixed algebraic closure Ω and ζn denote a primitive n-th root of unity ( n ≥ 1). Let k∞ be the maximal cyclotomic extension of k0, i.e. the field obtained by adjoining to k0 all ζn ( n = 1, 2, ...). Let M and L be the maximal abelian extension of k∞ and the maximal unramified abelian extension of k∞ respectively. The Galois groups Gal(M/k∞) and Gal...
متن کاملInterpretations and differential Galois extensions. DRAFT
We give model-theoretic accounts and proofs of the following results: Suppose ∂y = Ay is a linear differential equation over a differential field K of characteristic 0, and the field CK of constants of K is existentially closed in K. Then: (i) There exists a Picard-Vessiot extension L of K, namely a differential field extension L of K which is generated by a fundamental system of solutions of t...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Mathematical Logic Quarterly
سال: 2018
ISSN: 0942-5616
DOI: 10.1002/malq.201600098