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Reverse Mathematics and Algebraic Field Extensions
[article]

2013
*
arXiv
*
pre-print

This paper analyzes theorems about

arXiv:1209.4944v2
fatcat:4anp6sstufgxnfmcwlbpo5cfui
*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. ...##
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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. ...##
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Considerations on the hyperbolic complex Klein–Gordon equation

2010
*
Journal of Mathematical Physics
*

The hyperbolic complex numbers are applied in the sense that complex

doi:10.1063/1.3397456
fatcat:dfziazqqdbcq5ie2etgc54hu2q
*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 . ...##
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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 ( ...##
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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. ...##
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Solvability of Polynomials and Galois Group

2013
*
IOSR Journal of Mathematics
*

That is "if a polynomial is solvable by radicals, then the automorphism group of its splitting

doi:10.9790/5728-0512930
fatcat:2bjoungbyrdlxlomd3epgve7n4
*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. ...##
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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 ...##
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A computer algebra approach to biological systems

2003
*
Proceedings of the 2003 international symposium on Symbolic and algebraic computation - ISSAC '03
*

This paper focuses on dynamic networks over finite

doi:10.1145/860854.860859
dblp:conf/issac/Laubenbacher03
fatcat:b3e7ylenrzh7zoatzibthwm44e
*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. ...##
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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. ...##
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Formalizing Galois Theory
[article]

2021
*
arXiv
*
pre-print

The main theorems we formalized are the primitive element theorem, the fundamental theorem of Galois theory,

arXiv:2107.10988v1
fatcat:y6vfekokdzbc7jfmgwv5ui7iwm
*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. ...##
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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. ...##
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Book Review: Arithmetical questions on algebraic varieties

1952
*
Bulletin of the American Mathematical Society
*

one (the familiar figure of 27 lines) occurs in an

doi:10.1090/s0002-9904-1952-09625-7
fatcat:n5grwamq5zchbm7kq3muay4ex4
*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." ...##
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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. ...##
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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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