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p-Algebras of Arbitrary Exponents
1State University of New York at Buffalo, Buffalo, New York 14221.
Summary
This study introduces a differential polynomial ring associated with a Galois field extension. This construction method generates central simple algebras containing the extension as a maximal subalgebra using Witt vectors.
Area of Science:
- Algebraic Number Theory
- Commutative Algebra
- Galois Theory
Background:
- Galois field extensions are fundamental in algebra.
- Central simple algebras are key objects in non-commutative algebra.
- Maximal commutative subalgebras play a crucial role in understanding algebra structure.
Purpose of the Study:
- To associate a differential polynomial ring with a purely inseparable Galois field extension.
- To demonstrate a method for constructing central simple algebras.
- To explore the relationship between differential polynomial rings, Witt vectors, and central simple algebras.
Main Methods:
- Construction of a differential polynomial ring D.
- Association of D with a purely inseparable Galois field extension C over A.
- Factoring out specific ideals determined by Witt vectors from D.
Main Results:
- The study shows that all central simple A-algebras containing C as a maximal commutative subalgebra can be obtained from D.
- The construction relies on ideals determined by Witt vectors.
- This provides a novel approach to generating these algebras.
Conclusions:
- The differential polynomial ring construction offers a new perspective on central simple algebras.
- Witt vectors are integral to the factorization process.
- This work connects Galois theory, differential algebra, and non-commutative algebra.