[hal-00137158, v1] Well known theorems on triangular systems and
... of A′ then φ−1 (πq′1 ) ∩ · · · ∩ φ−1 (πq′r ) is a minimal primary decomposition of A. Proof. We use the notations of extensions and contractions defined in [24, chapter IV, paragraph 8], w.r.t. the ring homomorphism φ so that (φA) = Ae . The ideal πA′ is equal to the ideal Ae since both ideals admit ...
... of A′ then φ−1 (πq′1 ) ∩ · · · ∩ φ−1 (πq′r ) is a minimal primary decomposition of A. Proof. We use the notations of extensions and contractions defined in [24, chapter IV, paragraph 8], w.r.t. the ring homomorphism φ so that (φA) = Ae . The ideal πA′ is equal to the ideal Ae since both ideals admit ...
solution
... Thus, the inverse of 19 modulo 141 is 52. (b) Solve the congruence 19x ≡ 7 (mod 141), by specifying all the integer solutions x that satisfy the congruence. We have that the inverse of 19 modulo 141 is 52, so we can multiply both sides of the equation by 52: 19x ≡ 7 (mod 141) 52 · 19x ≡ 52 · 7 (mod ...
... Thus, the inverse of 19 modulo 141 is 52. (b) Solve the congruence 19x ≡ 7 (mod 141), by specifying all the integer solutions x that satisfy the congruence. We have that the inverse of 19 modulo 141 is 52, so we can multiply both sides of the equation by 52: 19x ≡ 7 (mod 141) 52 · 19x ≡ 52 · 7 (mod ...
Lecture 3
... An integer p > 1 is called a prime if it is not divisible by any integer other than 1, −1, p and −p. Thus p > 1 is a prime if it cannot be written as the product of two smaller positive integers. An integer n > 1 that is not a prime is called composite (the number 1 is considered neither prime, nor ...
... An integer p > 1 is called a prime if it is not divisible by any integer other than 1, −1, p and −p. Thus p > 1 is a prime if it cannot be written as the product of two smaller positive integers. An integer n > 1 that is not a prime is called composite (the number 1 is considered neither prime, nor ...
of integers satisfying a linear recursion relation
... From (5.1), we see that (W)n admits all the periods of (T)n, so that p \t. If (W)n is also non-singular, a repetition of the argument shows that r |p, so that T=p. The characteristic number r is called the principal period of (1.1) modulo m. It is easily shown that if (m, c) = 1, the sequences (W)n ...
... From (5.1), we see that (W)n admits all the periods of (T)n, so that p \t. If (W)n is also non-singular, a repetition of the argument shows that r |p, so that T=p. The characteristic number r is called the principal period of (1.1) modulo m. It is easily shown that if (m, c) = 1, the sequences (W)n ...
Continued Fractions in Approximation and Number Theory
... been able to discover, the bound on the period given in Theorems 9.4 and 10.1 is an original result, although admittedly a minor one. Theorems 12.2 to 12.4 (from Perron [8]) are significant and fairly deep results, rarely found in discussions of continued fractions or the Pell ...
... been able to discover, the bound on the period given in Theorems 9.4 and 10.1 is an original result, although admittedly a minor one. Theorems 12.2 to 12.4 (from Perron [8]) are significant and fairly deep results, rarely found in discussions of continued fractions or the Pell ...
Five, Six, and Seven-Term Karatsuba
... GFðpm Þ are important to cryptography; when p is prime and m > 1, field elements are represented by polynomials over the base field GFðpÞ. Computer algebra systems manipulate high and low-degree polynomials, often in multiple variables. Polynomial addition and subtraction algorithms have little inte ...
... GFðpm Þ are important to cryptography; when p is prime and m > 1, field elements are represented by polynomials over the base field GFðpÞ. Computer algebra systems manipulate high and low-degree polynomials, often in multiple variables. Polynomial addition and subtraction algorithms have little inte ...
1 Introduction 2 Integer Division
... Proof. These cells are also called equivalence classes. To see that two cells [u] and [v] are identical if have any integers in common, note that if w belongs to both [u] and [v], then u ≡ w (mod m) and v ≡ w (mod m), by the definition of cell. By symmetry w ≡ v (mod m) and by transitivity u ≡ v (mod ...
... Proof. These cells are also called equivalence classes. To see that two cells [u] and [v] are identical if have any integers in common, note that if w belongs to both [u] and [v], then u ≡ w (mod m) and v ≡ w (mod m), by the definition of cell. By symmetry w ≡ v (mod m) and by transitivity u ≡ v (mod ...
Here
... Factor Theorem: Suppose that P (x) is a polynomial. Then r is a root of P (x) if and only if P (x) = (x − r)Q(x) for some polynomial Q(x); in other words, if and only if (x − r) is a factor of P (x). Example: Consider the polynomial x3 − 4x2 − 15x + 18. We know that 1 is a root of this polynomial be ...
... Factor Theorem: Suppose that P (x) is a polynomial. Then r is a root of P (x) if and only if P (x) = (x − r)Q(x) for some polynomial Q(x); in other words, if and only if (x − r) is a factor of P (x). Example: Consider the polynomial x3 − 4x2 − 15x + 18. We know that 1 is a root of this polynomial be ...
37(2)
... this would imply that there exists a balancing number B between B0 and Bx, which is false. Thus, Hn is true for n = 1,2,.... This completes the proof of Theorem 3.1. Corollary 3.2: If x is any balancing number, then its previous balancing number is 3x - V8x2 + 1 . Proof: G(3x-V8x 2 + l) = x. ...
... this would imply that there exists a balancing number B between B0 and Bx, which is false. Thus, Hn is true for n = 1,2,.... This completes the proof of Theorem 3.1. Corollary 3.2: If x is any balancing number, then its previous balancing number is 3x - V8x2 + 1 . Proof: G(3x-V8x 2 + l) = x. ...
Lecture Notes for Chap 6
... it would have a linear factor x or x + 1; i.e., 0 or 1 would be a root of g(x), but g(0) = g(1) = 1 ∈ Z2 . 3). Using the same arguments as in 2), we can show that both 1 + x + x3 and 1 + x2 + x3 are irreducible over Z2 as they have no linear factors. Definition 6.2 (Division Rule). Let f (x) ∈ F [x] ...
... it would have a linear factor x or x + 1; i.e., 0 or 1 would be a root of g(x), but g(0) = g(1) = 1 ∈ Z2 . 3). Using the same arguments as in 2), we can show that both 1 + x + x3 and 1 + x2 + x3 are irreducible over Z2 as they have no linear factors. Definition 6.2 (Division Rule). Let f (x) ∈ F [x] ...
A SIMPLE PROOF OF SOME GENERALIZED PRINCIPAL IDEAL
... minimal primes, as required for the first statement. The second statement follows at once. 3. Heights of determinantal ideals The formulas for the heights of determinantal ideals that follow from Theorem 2.2 are often sharp. But under an additional hypothesis on M , Kwieciński ([8, Theorem 1]) gave ...
... minimal primes, as required for the first statement. The second statement follows at once. 3. Heights of determinantal ideals The formulas for the heights of determinantal ideals that follow from Theorem 2.2 are often sharp. But under an additional hypothesis on M , Kwieciński ([8, Theorem 1]) gave ...
Polynomials for MATH136 Part A
... The Fundamental Theorem states that not only complex quadratics but complex cubics, complex quartics, ... have complex roots. Because of this we say that C is algebraically closed. No new numbers need to be invented. The history of number consisted of mathematicians continuing to extend the number s ...
... The Fundamental Theorem states that not only complex quadratics but complex cubics, complex quartics, ... have complex roots. Because of this we say that C is algebraically closed. No new numbers need to be invented. The history of number consisted of mathematicians continuing to extend the number s ...
A.2 Polynomial Algebra over Fields
... that axi · bxj = (ab)xi+j always. (As usual we shall omit the · in multiplication when convenient.) The set F [x] equipped with the operations + and · is the polynomial ring in x over the field F . F is the field of coefficients of F [x]. Polynomial rings over fields have many of the properties enjo ...
... that axi · bxj = (ab)xi+j always. (As usual we shall omit the · in multiplication when convenient.) The set F [x] equipped with the operations + and · is the polynomial ring in x over the field F . F is the field of coefficients of F [x]. Polynomial rings over fields have many of the properties enjo ...
enumerating polynomials over finite fields
... We notice that the answer will always be a sum (±) of powers of q, where each power divides n, all divided by n. This means that as q gets large, the leading term, q n , will dominate to give cn ≈ q n /n. Thus for large q, the proportion of polynomials of degree n that are irreducible is approximate ...
... We notice that the answer will always be a sum (±) of powers of q, where each power divides n, all divided by n. This means that as q gets large, the leading term, q n , will dominate to give cn ≈ q n /n. Thus for large q, the proportion of polynomials of degree n that are irreducible is approximate ...
Dividing Polynomials
... dividend with like terms lined up. • Subtract the product from the dividend. • Bring down the next term in the original dividend and write it next to the remainder to form a new dividend. • Use this new expression as the dividend and repeat this process until the remainder can no longer be divided. ...
... dividend with like terms lined up. • Subtract the product from the dividend. • Bring down the next term in the original dividend and write it next to the remainder to form a new dividend. • Use this new expression as the dividend and repeat this process until the remainder can no longer be divided. ...
SPECIAL PRIME NUMBERS AND DISCRETE LOGS IN FINITE
... al b ≡ q1 q2 · · · qr (mod p) for prime integers qi ≤ p1/k . Next the logarithm of each qi must be evaluated. For this purpose, authors of [3, 5] sieve the values of polynomials f (x) = fqi (x) dependent on qi for which (1) holds. The advantage of our method is that congruence (2) or (1) does not de ...
... al b ≡ q1 q2 · · · qr (mod p) for prime integers qi ≤ p1/k . Next the logarithm of each qi must be evaluated. For this purpose, authors of [3, 5] sieve the values of polynomials f (x) = fqi (x) dependent on qi for which (1) holds. The advantage of our method is that congruence (2) or (1) does not de ...
Explicit formulas for Hecke Gauss sums in quadratic
... follows from (2). It is remarkable, that hence, as a consequence, the quadratic reciprocity law for quadratic number fields is not a genuine new reciprocity law. This is in contrast to what is suggested by Hecke’s proof which makes extensive use of theta series associated to number fields. A formula ...
... follows from (2). It is remarkable, that hence, as a consequence, the quadratic reciprocity law for quadratic number fields is not a genuine new reciprocity law. This is in contrast to what is suggested by Hecke’s proof which makes extensive use of theta series associated to number fields. A formula ...
Full text
... the first of which uses the least absolute remainder at each step and which is shorter than the others. A theorem of Kronecker, see Uspensky & Heaslet [3], says that no Euclidean algorithm is shorter than the one obtained by taking the least absolute remainder at each step of division. Goodman & Zar ...
... the first of which uses the least absolute remainder at each step and which is shorter than the others. A theorem of Kronecker, see Uspensky & Heaslet [3], says that no Euclidean algorithm is shorter than the one obtained by taking the least absolute remainder at each step of division. Goodman & Zar ...
Z/mZ AS A NUMBER SYSTEM As useful as the congruence notation
... This awkwardness is addressed by introducing the generalized number system Z/mZ. Congruence classes. A preliminary definition is: Definition 2. Given a ∈ Z, the congruence class of a modulo m, denoted [a]m ,1 is the subset of Z consisting of all integers modulo m. A congruence class modulo m is a su ...
... This awkwardness is addressed by introducing the generalized number system Z/mZ. Congruence classes. A preliminary definition is: Definition 2. Given a ∈ Z, the congruence class of a modulo m, denoted [a]m ,1 is the subset of Z consisting of all integers modulo m. A congruence class modulo m is a su ...
An Approach to Hensel`s Lemma
... sometimes all that is required; for example, this occurs in the lifting of generator polynomials of cyclic codes over Z2 to Z4 , or more generally from Zp to Zpk . Coding theorists have been using the method of this article, and although we have not seen anything in print, this article contains noth ...
... sometimes all that is required; for example, this occurs in the lifting of generator polynomials of cyclic codes over Z2 to Z4 , or more generally from Zp to Zpk . Coding theorists have been using the method of this article, and although we have not seen anything in print, this article contains noth ...
Solutions - Math@LSU
... matter. That is, if x −→ y then y −→ x, because if we have x −→ y −→ z, then xy = n = yz so x = z. So this relation pairs the divisors of n, and since all the divisors are paired uniquely, the parity of the number of elements in the set is equal to the parity of the number of elements that are mappe ...
... matter. That is, if x −→ y then y −→ x, because if we have x −→ y −→ z, then xy = n = yz so x = z. So this relation pairs the divisors of n, and since all the divisors are paired uniquely, the parity of the number of elements in the set is equal to the parity of the number of elements that are mappe ...
Document
... Divide the first term in the dividend by the first term in the divisor. The result is the first term of the quotient. Multiply every term in the divisor by the first term in the quotient. Write the resulting product beneath the dividend with like terms lined up. Subtract the product from the dividen ...
... Divide the first term in the dividend by the first term in the divisor. The result is the first term of the quotient. Multiply every term in the divisor by the first term in the quotient. Write the resulting product beneath the dividend with like terms lined up. Subtract the product from the dividen ...
1 Homework 1
... SOLUTION: Suppose x, y ∈ rad(I) and xn , y m ∈ I. Then binomial expansion of (x + y)n+m−1 shows that each term is either of degree at least n in x or degree at least m in y, hence (x + y) ∈ rad(I). For the multiplicative property, xr ∈ rad(I) since (xr)n = xn rn ∈ I since xn ∈ I. (3) Factor 1 + 3i i ...
... SOLUTION: Suppose x, y ∈ rad(I) and xn , y m ∈ I. Then binomial expansion of (x + y)n+m−1 shows that each term is either of degree at least n in x or degree at least m in y, hence (x + y) ∈ rad(I). For the multiplicative property, xr ∈ rad(I) since (xr)n = xn rn ∈ I since xn ∈ I. (3) Factor 1 + 3i i ...
The Repeated Sums of Integers
... for me not only for the course of my thesis research, but also for my whole journey in Mathematics. Dr. Mihaila was the one to show me that I have the potential to pursue a career in Math. I would also like to thank all of my Math professors. Each and everyone of you have shown me a different side of ...
... for me not only for the course of my thesis research, but also for my whole journey in Mathematics. Dr. Mihaila was the one to show me that I have the potential to pursue a career in Math. I would also like to thank all of my Math professors. Each and everyone of you have shown me a different side of ...
1 Lecture 1
... Undefined terms: set of numbers N = {1, 2, 3, ...} with the property that if n, m ∈ N then n + m ∈ N and nm ∈ N. 1. Axiom 1 1 belongs to N : there is a number 1 such that 1 ∗ n ∈ N for any n ∈ N. 2. Axiom2 There is no number n such that n + 1 = 1. 3. Axiom 3 If S is a subset of N with the property t ...
... Undefined terms: set of numbers N = {1, 2, 3, ...} with the property that if n, m ∈ N then n + m ∈ N and nm ∈ N. 1. Axiom 1 1 belongs to N : there is a number 1 such that 1 ∗ n ∈ N for any n ∈ N. 2. Axiom2 There is no number n such that n + 1 = 1. 3. Axiom 3 If S is a subset of N with the property t ...
Chinese remainder theorem
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra. It was first published in the 3rd to 5th centuries by the Chinese mathematician Sun Tzu.In its basic form, the Chinese remainder theorem will determine a number n that, when divided by some given divisors, leaves given remainders. For example, what is the lowest number n that when divided by 3 leaves a remainder of 2, when divided by 5 leaves a remainder of 3, and when divided by 7 leaves a remainder of 2?