
Chap 6
... In other words, no matter which elements in each sector we choose, the sums and products are equivalent. This allows us to compute sums and products by choosing any representatives we want. Example 6.2. Referring to the equivalence classes for m = 5 of Example 6.1 and the letters of Theorem 5.3, tak ...
... In other words, no matter which elements in each sector we choose, the sums and products are equivalent. This allows us to compute sums and products by choosing any representatives we want. Example 6.2. Referring to the equivalence classes for m = 5 of Example 6.1 and the letters of Theorem 5.3, tak ...
STRUCTURE THEOREMS OVER POLYNOMIAL RINGS 1
... Condition (1) of the theorem is independent of the ring R, so if, for given S, k and G, one of the other conditions is satisfied for some ring R then it is also satisfied by any other ring R satisfying the hypotheses of the theorem. We thank the referee for the elegant proofs of 3.3 and 8.1. 2. Chan ...
... Condition (1) of the theorem is independent of the ring R, so if, for given S, k and G, one of the other conditions is satisfied for some ring R then it is also satisfied by any other ring R satisfying the hypotheses of the theorem. We thank the referee for the elegant proofs of 3.3 and 8.1. 2. Chan ...
Full text
... testing if a number is representable as a sum of three cubes, nor a method for finding such a representation (see [12]). For the quadratic forms, the situation is different; Gauss developed his theory of quadratic forms [7] and solved the related integer representation problem. In particular, quadra ...
... testing if a number is representable as a sum of three cubes, nor a method for finding such a representation (see [12]). For the quadratic forms, the situation is different; Gauss developed his theory of quadratic forms [7] and solved the related integer representation problem. In particular, quadra ...
DEGREE OF REGULARITY FOR HFE
... consisting of polynomials p1 (x1 , . . . , xn ), . . . , pn (x1 , . . . , xn ) is the lowest degree at which we have non-trivial polynomial relations between the pi components. It is commonly accepted that in general this is the degree at which the solving algorithm will terminate and therefore it i ...
... consisting of polynomials p1 (x1 , . . . , xn ), . . . , pn (x1 , . . . , xn ) is the lowest degree at which we have non-trivial polynomial relations between the pi components. It is commonly accepted that in general this is the degree at which the solving algorithm will terminate and therefore it i ...
Basic Algorithmic Number Theory
... We give general definitions for the success probability of an algorithm in this section, but rarely use the formalism in our later discussion. Instead, for most of the book, we focus on the case of algorithms that always succeed (or, at least, that succeed with probability extremely close to 1). Thi ...
... We give general definitions for the success probability of an algorithm in this section, but rarely use the formalism in our later discussion. Instead, for most of the book, we focus on the case of algorithms that always succeed (or, at least, that succeed with probability extremely close to 1). Thi ...
Homework #5 Solutions (due 10/10/06)
... Yet another way of expressing this is that NG may be regarded as a function on the set of conjugacy classes of subgroups. Now we note that almost all of our subgroups can be identified as either cyclic subgroups or as certain normalizers (or centralizers). Cyclic subgroups are easily divided into c ...
... Yet another way of expressing this is that NG may be regarded as a function on the set of conjugacy classes of subgroups. Now we note that almost all of our subgroups can be identified as either cyclic subgroups or as certain normalizers (or centralizers). Cyclic subgroups are easily divided into c ...
Part XV Appendix to IO54
... (2) The group (G , ◦) is called cyclic if it contains an element g , called generator, such that the order of (g ) = |G |. Theorem If the multiplicative group (Zn? , ×n ) is cyclic, then it is isomorphic to the additive group (ZΦ(n) , +Φ(n) ). (However, no effective way is known, given n, to create ...
... (2) The group (G , ◦) is called cyclic if it contains an element g , called generator, such that the order of (g ) = |G |. Theorem If the multiplicative group (Zn? , ×n ) is cyclic, then it is isomorphic to the additive group (ZΦ(n) , +Φ(n) ). (However, no effective way is known, given n, to create ...
the arithmetical theory of linear recurring series
... is self-explanatory. The following convenient definition was introduced by H. T. Engstrom*: A number ir is said to be a general period of the difference equation (1.1) for the modulus m if every sequence of rational integers (m) satisfying (1.1) has the period ir. Let t be the least such general per ...
... is self-explanatory. The following convenient definition was introduced by H. T. Engstrom*: A number ir is said to be a general period of the difference equation (1.1) for the modulus m if every sequence of rational integers (m) satisfying (1.1) has the period ir. Let t be the least such general per ...
(pdf)
... Example 2.20. Recalling the algebraic integer 2 + 3 = α from example 2.8, we compute NQ (α). This amounts to taking the determinant of the linear transformation in 2.8, or, because we are lazy, simply plugging in 0 for the minimal polynomial of α in the determinant already computed. Then N (α) = 1, ...
... Example 2.20. Recalling the algebraic integer 2 + 3 = α from example 2.8, we compute NQ (α). This amounts to taking the determinant of the linear transformation in 2.8, or, because we are lazy, simply plugging in 0 for the minimal polynomial of α in the determinant already computed. Then N (α) = 1, ...
Section 0. Background Material in Algebra, Number Theory and
... numbers are in the same coset iff they have the same modulus. Clearly, the left cosets are just the circles with centre 0, and these are the elements of G/H. We have ‘modded out’ by H, removing the argument information, and retaining only the modulus information. We can turn G/H into a group under m ...
... numbers are in the same coset iff they have the same modulus. Clearly, the left cosets are just the circles with centre 0, and these are the elements of G/H. We have ‘modded out’ by H, removing the argument information, and retaining only the modulus information. We can turn G/H into a group under m ...
2.1, 2.3-2.5 Review
... An asset with a first cost of $100,000 is depreciated over 5-year period. It is expected to have a $10,000 salvage value at the end of 5 years. Using the straight-line method, what is the book value at the end of year 2? ...
... An asset with a first cost of $100,000 is depreciated over 5-year period. It is expected to have a $10,000 salvage value at the end of 5 years. Using the straight-line method, what is the book value at the end of year 2? ...
EIGENVALUES OF PARTIALLY PRESCRIBED
... This paper is a natural generalization of those results. As the main result (Theorem 3.1), we give a complete solution of Problem 1.1 in the case when the eigenvalues of the matrix (1.2) belong to F, and F is an infinite field. In particular, this gives the complete solution of Problem 1.1 over algebr ...
... This paper is a natural generalization of those results. As the main result (Theorem 3.1), we give a complete solution of Problem 1.1 in the case when the eigenvalues of the matrix (1.2) belong to F, and F is an infinite field. In particular, this gives the complete solution of Problem 1.1 over algebr ...