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Reverse Mathematics and Algebraic Field Extensions [article]

François G. Dorais, Jeffry Hirst, Paul Shafer
2013 arXiv   pre-print
This paper analyzes theorems about algebraic field extensions using the techniques of reverse mathematics.  ...  Normal and Galois extensions are discussed in section 4, and the Galois correspondence theorems for infinite field extensions are treated in section 5.  ...  For every field F and every algebraic extension K of F , NOR4 → NOR1. For every field F and every algebraic extension K of F , NOR4 → NOR3. 4.  ... 
arXiv:1209.4944v2 fatcat:4anp6sstufgxnfmcwlbpo5cfui

Page 697 of Mathematical Reviews Vol. 37, Issue 3 [page]

1969 Mathematical Reviews  
Authors’ summary: “The general algebraic relations be- tween space inversion, time reversal, and the internal symmetry group are analyzed within the framework of a Lorentz-invariant local field theory.  ...  C. 3816 Space inversion, time reversal, and other discrete symmetries in local field theories. Phys. Rev. (2) 148 (1966), 1385-1404.  ... 

Considerations on the hyperbolic complex Klein–Gordon equation

S. Ulrych
2010 Journal of Mathematical Physics  
The hyperbolic complex numbers are applied in the sense that complex extensions of groups and algebras are performed not with the complex unit, but with the product of complex and hyperbolic unit.  ...  The article summarizes and consolidates investigations on hyperbolic complex numbers with respect to the Klein-Gordon equation for fermions and bosons.  ...  The terminology of reversion and conjugation refers to the Clifford algebra approach. The hyperbolic Pauli algebra corresponds to the real Clifford algebra R 3,0 .  ... 
doi:10.1063/1.3397456 fatcat:dfziazqqdbcq5ie2etgc54hu2q

Page 6049 of Mathematical Reviews Vol. , Issue 2004h [page]

2004 Mathematical Reviews  
For finite groups F, G and a field k, let AF denote the group Hopf algebra, and let k° denote the dual Hopf algebra of kG. Essentially by G. I.  ...  The authors prove various facts about reversible rings and their extensions; for example, they show that a reversible ring is semicommutative (i.e. every annihilator is an ideal), and if R is reduced (  ... 

Page 9289 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
The relation between some infinite dimensional loop algebras, such as the Virasoro-like algebra, and the Euclidean algebras e(2) and e(3) is also analyzed.” 2004k:81107 81Q05 Calogero, F.  ...  They represent possible self-adjoint extensions of the nonrelativistic kinetic-energy operator. Assuming time-reversal invariance we find a family of self-adjoint extensions with seven parameters.  ... 

Solvability of Polynomials and Galois Group

YahayaShagaiya Daniel
2013 IOSR Journal of Mathematics  
That is "if a polynomial is solvable by radicals, then the automorphism group of its splitting field must be a solvable group." Field theory is connected with Group theory.  ...  It has been found that Galois Theory can be used to determine the solvability of polynomials over a field by radicals.  ...  The general fifth degree polynomial equation in one indeterminate is not solvable by radicals over the field of rational numbers and polynomial equations of degree≤ 4 are solvable by radicals.  ... 
doi:10.9790/5728-0512930 fatcat:2bjoungbyrdlxlomd3epgve7n4

Page 832 of American Journal of Mathematics Vol. 79, Issue 4 [page]

1957 American Journal of Mathematics  
Also (11) given a one-dimensional algebraic function field L/k and a finite separable algebraic extension L* of L there exists a finite algebraic  ...  Then (1) given a one-dimensional algebraic function field L/k there exists a finite algebraic extension L, of K such that L, is k-isomorphi to L and such that v is the only valuation of K/k which is possibly  ... 

A computer algebra approach to biological systems

Reinhard Laubenbacher
2003 Proceedings of the 2003 international symposium on Symbolic and algebraic computation - ISSAC '03  
This paper focuses on dynamic networks over finite fields and applications to the modeling and analysis of biological networks using tools from computer algebra, in particular gene regulatory networks,  ...  and agent-based simulations of processes in computational immunology.  ...  Jarrah, Pedro Mendes, Michael Stillman, and Bernd Sturmfels.  ... 
doi:10.1145/860854.860859 dblp:conf/issac/Laubenbacher03 fatcat:b3e7ylenrzh7zoatzibthwm44e

Page 575 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 58, Issue 5 [page]

1952 American Mathematical Society. Bulletin of the American Mathematical Society  
notions and results have played a réle in algebraic geometry.”  ...  point, and every line through this point is a tangent, though its point of contact may not belong to y but to a quadratic extension of y.  ... 

Formalizing Galois Theory [article]

Thomas Browning, Patrick Lutz
2021 arXiv   pre-print
The main theorems we formalized are the primitive element theorem, the fundamental theorem of Galois theory, and the equivalence of several characterizations of finite degree Galois extensions.  ...  We discuss some of the challenges we faced and the decisions we made in the course of this project.  ...  field theory in mathlib on which our project depended.  ... 
arXiv:2107.10988v1 fatcat:y6vfekokdzbc7jfmgwv5ui7iwm

Page 71 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 82, Issue 1 [page]

1956 American Mathematical Society. Transactions of the American Mathematical Society  
Let Z be a general cycle of M/ko and let AC|Z| Lins. Then ko(A) 4s a finite separable algebraic extension of ko(Z). Proof.  ...  Reversing the procedure enables us to conclude that A and A’ are conjugate points over the field ko(Z). There- fore ® is independent of the particular point chosen from | Z| OL‘,-s. LemMA 8.  ... 

Book Review: Arithmetical questions on algebraic varieties

Patrick DuVal
1952 Bulletin of the American Mathematical Society  
one (the familiar figure of 27 lines) occurs in an algebraically closed field, and only four in the real field, but all ten in the rational field.  ...  notions and results have played a rôle in algebraic geometry."  ... 
doi:10.1090/s0002-9904-1952-09625-7 fatcat:n5grwamq5zchbm7kq3muay4ex4

Page 5815 of Mathematical Reviews Vol. , Issue 2002H [page]

2002 Mathematical Reviews  
of mathematics and physics.  ...  In the present paper, two similar results are proved, one for real von Neumann algebras, and one for “reversible” Jordan-von Neumann algebras.  ... 

Page 409 of Illinois Journal of Mathematics Vol. 5, Issue 3 [page]

1961 Illinois Journal of Mathematics  
ZASsSENHAUS We shall use the following notations: K = algebraic number field, R = valuation ring in K with maximal ideal P, K’ = finite extension field over K, R’ = valuation ring of K’ containing R; A  ...  EQUIVALENCE OF REPRESENTATIONS UNDER EXTENSIONS OF LOCAL GROUND RINGS! BY I. REINER AND H.  ... 

Page 8746 of Mathematical Reviews Vol. , Issue 2001M [page]

2001 Mathematical Reviews  
The authors reverse in this work this process and begin with the Dynkin diagram defining the Lie algebra.  ...  This algebraic notion generalises that of an ideal (when @ is injective) and that of a central extension (when @ is surjective).  ... 
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