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On maximal orders of division quaternion algebras over the rational number field with certain optimal embeddings
1982
Nagoya mathematical journal
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In this paper, we shall give explicitZ-basis of certain maximal orders of definite quaternion algebras over the rational number fieldQ(See Theorems below). ...
We shall also give some remarks on symmetric maximal orders in Ponomarev [9] and Hashimoto [6] (Proposition 4.3). More precise contents are as follows. LetDbe a division quaternion algebra overQ. ...
Besides, when m = 3 mod 4, we choose a rational integer r r such that ( 4) r' 2 + m = 0 mod 4<? . Put θ = Z + Z-ii^-+ Z/3 + Z (r ' + a)β . 2 2g Then, this is also a maximal order of D. ...
doi:10.1017/s002776300002016x
fatcat:brw3uajhz5d4vbv3yc7fhanvjm
Page 171 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 13, Issue 2
[page]
1912
American Mathematical Society. Transactions of the American Mathematical Society
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Now suppose, if possible, that there exists an integral rational function G, (2, v) which is not divisible by z — a, and which possesses y,, uw, -**, H, as its orders of coincidence with the branches of ...
By repeated applications of the above process, it would evidently be possible to construct an integral rational function G (z, 7), which is not divisible by z — a, and whose orders of coincidence with ...
Page 4 of The Month Vol. 28, Issue 148
[page]
1876
The Month
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Defi- nition of Philosophy. Divisions. Logic. Zhe First Part.—On the efficient cause of the Rational Order. Lecture III. The Efficient Cause of the Rational Order. Zhe Second Part. ...
—On the material cause of the Rational Order. Lecture IV. Definitions and Divisions of Terms. Lecture V. On Definition and Division. Lecture VI. On the Definition and Division of Propositions. ...
c-Orderable division rings with involution
1982
Journal of Algebra
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Another important property of the c-ordering is that the set of bounded elements x at this ordering (e.g., XX* < rO for some rational r") is a *-valuation subring Y ...
In fact, any c-orderable division ring R is shown to admit an ordering of the following type. ...
In fact, any c-orderable division ring R is shown to admit an ordering of the following type. ...
doi:10.1016/0021-8693(82)90053-9
fatcat:5zmz3pjtcbcg3jjfjdbit4hopi
Divisibility Monoids: Presentation, Word Problem, and Rational Languages
[chapter]
2001
Lecture Notes in Computer Science
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Essentially, Theorem 13 says that a divisibility monoid satisfies Kleene's Theorem if and only if it is rational if and only if it is width-bounded. Thus, in the context of divisibility monoids, ...
It turns out that this is the case iff the divisibility monoid is a rational monoid [25] iff it is width-bounded. ...
Ordered monoids where the order relation is the intersection of the prefix and the suffix relation were investigated e.g. in [2] under the name "divisibility monoid". ...
doi:10.1007/3-540-44669-9_23
fatcat:t3e5knp7bvfbxowdzhurbfxxhq
Quaternion constituents of group algebras
1971
Proceedings of the American Mathematical Society
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In this paper it is shown that each quaternion division algebra central over the rationals appears as a division ring constituent of some rational group algebra. ...
For rational division algebras, the Hasse invariants correspond to the set of rational primes including the infinite prime. ...
Let A be a cyclic group of order p, T cyclic of order 2n+1, and M cyclic of order m. Set G = A(TXM), the semidirect product with A <JG and |GTAfG4)|=2. ...
doi:10.1090/s0002-9939-1971-0280609-0
fatcat:gc5oixk6kjdaxmfgqoktrmixxa
The Cyclic Group as a Basic Element in the Theory of Numbers
1911
The American mathematical monthly
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is divisible by the rational integer m. ...
The term order of a rational integer a with respect to modulus m, m being a positive rational integer, will be used for the smallest positive rational integer b which is such that the product ab is divisible ...
doi:10.1080/00029890.1911.11997639
fatcat:266vnpitq5dgxhjjinmvxvbkay
The Cyclic Group as a Basic Element in the Theory of Numbers
1911
The American mathematical monthly
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The term order of a rational integer a with respect to modulus m, mn being a positive rational integer, will be used for the smallest positive rational integer b which is such that the product ab is divisible ...
This proves the theorem: If paiS the highest power of a rational prime number which divides the order (mod m) of one of two rational integers without dividing this order of the other, then the order (mod ...
doi:10.2307/2972576
fatcat:ddoferkblvaubo7i3fmvopyope
Page 493 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 97, Issue 3
[page]
1960
American Mathematical Society. Transactions of the American Mathematical Society
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Clearly the character group of X contains no elements of finite order since X is divisible. ...
Since H’ CH, H(\K =0, K has no nontrivial elements of finite order and consequently K is a vector space over the rational numbers [4, p. 10]. ...
ON NORMAL SUBGROUPS OF D∗ WHOSE ELEMENTS ARE PERIODIC MODULO THE CENTER OF D∗ OF BOUNDED ORDER
2014
International Electronic Journal of Algebra
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Let D be a division ring with the center F = Z(D). ...
Suppose that N is a normal subgroup of D * which is radical over F , that is, for any element x ∈ N , there exists a positive integer nx, such that x nx ∈ F . ...
ON NORMAL SUBGROUPS OF D * ...
doi:10.24330/ieja.266227
fatcat:pbepn5a2kfct7osqadejhyp6va
Primitive Points on Elliptic Curves
[chapter]
2000
Springer Collected Works in Mathematics
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Our purpose here is to formulate an analogous conjecture on elliptic curves A, say defined over the rationals for concreteness. Let a be a rational point of infinite order. ...
Note that / divides the order of -4(F) if and only if 7 p has eigenvalue 1. Furthermore, A(F ) = Ker (7 -1). If y p = 1 then the index of (a) is divisible by /. ...
Our purpose here is to formulate an analogous conjecture on elliptic curves A, say defined over the rationals for concreteness. Let a be a rational point of infinite order. ...
doi:10.1007/978-1-4614-8710-4_19
fatcat:mf67jsvjdzev3bznyajaeu2pii
Primitive Points on Elliptic Curves
[chapter]
2000
Collected Papers Volume II
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Our purpose here is to formulate an analogous conjecture on elliptic curves A, say defined over the rationals for concreteness. Let a be a rational point of infinite order. ...
Note that / divides the order of -4(F) if and only if 7 p has eigenvalue 1. Furthermore, A(F ) = Ker (7 -1). If y p = 1 then the index of (a) is divisible by /. ...
Our purpose here is to formulate an analogous conjecture on elliptic curves A, say defined over the rationals for concreteness. Let a be a rational point of infinite order. ...
doi:10.1007/978-1-4612-2120-3_19
fatcat:zjjyj5hz5nbh5ccb6cd25r7mbq
Primitive points on elliptic curves
1977
Bulletin of the American Mathematical Society
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Our purpose here is to formulate an analogous conjecture on elliptic curves A, say defined over the rationals for concreteness. Let a be a rational point of infinite order. ...
Note that / divides the order of -4(F) if and only if 7 p has eigenvalue 1. Furthermore, A(F ) = Ker (7 -1). If y p = 1 then the index of (a) is divisible by /. ...
Our purpose here is to formulate an analogous conjecture on elliptic curves A, say defined over the rationals for concreteness. Let a be a rational point of infinite order. ...
doi:10.1090/s0002-9904-1977-14310-3
fatcat:wzc5iumcabdapoz7pm5t6xxypq
Page 621 of Annals of Mathematics Vol. 30, Issue
[page]
1928
Annals of Mathematics
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THE STRUCTURE OF ANY ALGEBRA WHICH IS A DIRECT PRODUCT OF RATIONAL GENERALIZED QUATERNION DIVISION ALGEBRAS.*
By A. ADRIAN ALBERT.
1. Introduction. ...
The direct products of two generalized quaternion division
algebras over R. We shall consider rational linear associative algebras, ,
that is algebras over R, the field of all rational numbers. ...
On Division Algebras
1921
Transactions of the American Mathematical Society
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If B is a subalgebra of order b in a division algebra A of order a, there exists a complex C of order c such that A = BC, a = 6c. ...
It is shown in the present paper that the Dickson algebra is the only noncommutative algebra of order 9 so that the only division algebras of order not greater than 9 are (i) the Dickson algebras of order ...
If B is a subalgebra of order b in a division algebra A of order a, there exists a complex C of order c such that A = BC, a = 6c. ...
doi:10.2307/1989011
fatcat:hpku6ograbck7g5x2m3pp6f2tu
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