Filters








418,638 Hits in 3.8 sec

On maximal orders of division quaternion algebras over the rational number field with certain optimal embeddings

Tomoyoshi Ibukiyama
1982 Nagoya mathematical journal  
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  
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  
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

M Chacron
1982 Journal of Algebra  
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]

Dietrich Kuske
2001 Lecture Notes in Computer Science  
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

Mark Benard
1971 Proceedings of the American Mathematical Society  
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

G. A. Miller
1911 The American mathematical monthly  
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

G. A. Miller
1911 The American mathematical monthly  
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  
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

Mai Hoang Bien
2014 International Electronic Journal of Algebra  
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]

Serge Lang
2000 Springer Collected Works in Mathematics  
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]

S. Lang, H. Trotter
2000 Collected Papers Volume II  
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

S. Lang, H. Trotter
1977 Bulletin of the American Mathematical Society  
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  
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

J. H. M. Wedderburn
1921 Transactions of the American Mathematical Society  
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
« Previous Showing results 1 — 15 out of 418,638 results