number fields
... with aj ∈ Z. Choose n as small as possible, so a0 6= 0. If v(α) < 0, then heuristically αn has an n-th order pole at v, while the other terms in the sum on the left have a lower order pole (the aj ’s don’t contribute polar data since v(aj ) ≥ 0 or aj = 0). Thus the whole sum on the left has a pole a ...
... with aj ∈ Z. Choose n as small as possible, so a0 6= 0. If v(α) < 0, then heuristically αn has an n-th order pole at v, while the other terms in the sum on the left have a lower order pole (the aj ’s don’t contribute polar data since v(aj ) ≥ 0 or aj = 0). Thus the whole sum on the left has a pole a ...
X10c
... • Sometimes easier to define an object in terms of itself. • This process is called recursion. – Sequences • {s0,s1,s2, …} defined by s0 = a and sn = 2sn-1 + b for constants a and b and n Z+ ...
... • Sometimes easier to define an object in terms of itself. • This process is called recursion. – Sequences • {s0,s1,s2, …} defined by s0 = a and sn = 2sn-1 + b for constants a and b and n Z+ ...
A.2 Polynomial Algebra over Fields
... other polynomial of F [x] does. Again like the integers, F [x] does satisfy the cancellation law and so is an integral domain. These last two claims follow from some results of independent interest. To prove them we first need a (once again familiar) definition. The degree of the polynomial a(x) of ...
... other polynomial of F [x] does. Again like the integers, F [x] does satisfy the cancellation law and so is an integral domain. These last two claims follow from some results of independent interest. To prove them we first need a (once again familiar) definition. The degree of the polynomial a(x) of ...
4.2 Every PID is a UFD
... Definition 4.2.1 A commutative ring R satisfies the ascending chain condition (ACC) on ideals if there is no infinite sequence of ideals in R in which each term properly contains the previous one. Thus if I1 ⊆ I 2 ⊆ I 3 ⊆ . . . is a chain of ideals in R, then there is some m for which Ik = Im for al ...
... Definition 4.2.1 A commutative ring R satisfies the ascending chain condition (ACC) on ideals if there is no infinite sequence of ideals in R in which each term properly contains the previous one. Thus if I1 ⊆ I 2 ⊆ I 3 ⊆ . . . is a chain of ideals in R, then there is some m for which Ik = Im for al ...
Efficient Diffie-Hellman Two Party Key Agreement
... following: given elements a and b Î G, find an integer x, 0 £ x £ n - 1, such that ax = b, provided that such an integer exists. For two primary reasons, a variety of groups have been proposed for cryptographic use. First, the operation in some groups may be easier to implement in software or in har ...
... following: given elements a and b Î G, find an integer x, 0 £ x £ n - 1, such that ax = b, provided that such an integer exists. For two primary reasons, a variety of groups have been proposed for cryptographic use. First, the operation in some groups may be easier to implement in software or in har ...
Rationality and the Tangent Function
... This paper aims to reveal the geometric and algebraic significance of the irrational nature of the numbers tan kπ/n. For example, unique factorization of Gaussian integers yields a very natural proof that the only rational values of tan kπ/n are 0 and ±1 (Corollary 1 in Section 2). In all, we give f ...
... This paper aims to reveal the geometric and algebraic significance of the irrational nature of the numbers tan kπ/n. For example, unique factorization of Gaussian integers yields a very natural proof that the only rational values of tan kπ/n are 0 and ±1 (Corollary 1 in Section 2). In all, we give f ...
Primes in quadratic fields
... we have the zero ideal (0) containing only the number zero, and the unit ideal (1), which is equal to R. If all ideals in R are principal ideals, then R is called a ‘principal ideal domain’. A principal ideal domain necessarily is a unique-factorization domain. An ideal A of R induces subdivision of ...
... we have the zero ideal (0) containing only the number zero, and the unit ideal (1), which is equal to R. If all ideals in R are principal ideals, then R is called a ‘principal ideal domain’. A principal ideal domain necessarily is a unique-factorization domain. An ideal A of R induces subdivision of ...
LOCAL FIELDS AND p-ADIC GROUPS In these notes, we follow [N
... Consider the field Q of rational numbers. There is the typical absolute value | · |, which we will also denote by | · |∞ , which is an archimedean absolute value. For an example of a non-archimedean absolute value, fix a prime number p, and define an absolute value | · |p on Q as follows. For a nonz ...
... Consider the field Q of rational numbers. There is the typical absolute value | · |, which we will also denote by | · |∞ , which is an archimedean absolute value. For an example of a non-archimedean absolute value, fix a prime number p, and define an absolute value | · |p on Q as follows. For a nonz ...
Aurifeuillian factorizations - American Mathematical Society
... found similar identities for every composite exponent n not divisible by 8. He showed that if n = N, 2N or 4N , with N odd, and if d is any squarefree divisor of N (where d is allowed to be negative when n = 4N ), then there exist polynomials Un,d (x), Vn,d (x) ∈ Z[x] such that ϕN (x) = UN,d (x)2 − ...
... found similar identities for every composite exponent n not divisible by 8. He showed that if n = N, 2N or 4N , with N odd, and if d is any squarefree divisor of N (where d is allowed to be negative when n = 4N ), then there exist polynomials Un,d (x), Vn,d (x) ∈ Z[x] such that ϕN (x) = UN,d (x)2 − ...
Ring Theory
... A2 Addition is commutative. r + s = s + r for all r, s ∈ R. A3 R contains an identity element for addition, denoted by 0R and called the zero element of R. r + 0R = 0R + r = r for all r ∈ R. A4 Every element of R has an inverse with respect to addition. (The additive inverse of r is often denoted by ...
... A2 Addition is commutative. r + s = s + r for all r, s ∈ R. A3 R contains an identity element for addition, denoted by 0R and called the zero element of R. r + 0R = 0R + r = r for all r ∈ R. A4 Every element of R has an inverse with respect to addition. (The additive inverse of r is often denoted by ...
Document
... • Since G is finite, the set {e,α,α2,…} is finite. At some point, there must be some repetition. • Let αk=αk+t be the first repetition. Then αt=e. This t is called the order of α, denoted by ord(α). ...
... • Since G is finite, the set {e,α,α2,…} is finite. At some point, there must be some repetition. • Let αk=αk+t be the first repetition. Then αt=e. This t is called the order of α, denoted by ord(α). ...
Holt CA Course 1
... is a constant because the number cannot change. An algebraic expression is an expression that contains at least one variable. For example, 1954 + a is an algebraic expression. ...
... is a constant because the number cannot change. An algebraic expression is an expression that contains at least one variable. For example, 1954 + a is an algebraic expression. ...
