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Reverse Mathematics and Algebraic Field Extensions
[article]
2013
arXiv
pre-print
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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
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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
2010
Journal of Mathematical Physics
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2010 pre-print arXiv:1006.5182v1 fatcat:bat4bqk2yjcdfig6n5vwub342a
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
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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
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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
2013
IOSR Journal of Mathematics
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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
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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
2003
Proceedings of the 2003 international symposium on Symbolic and algebraic computation - ISSAC '03
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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
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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]
2021
arXiv
pre-print
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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
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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
1952
Bulletin of the American Mathematical Society
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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
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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
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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
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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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