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If R is a unique factorization domain and if a and b in R are relatively prime (i.e.,(a,b) = 1), whenever a divides bc, then a divides c.
That is, if R is a unique factorization domain and if a and b in R are relatively prime
(i.e., (a,b) = 1), whenever a divides bc then a divides c.
132134 If R is a unique factorization domain and if a and b in R are relatively prime (i.e.,(a,b) = 1), whenever a divides bc, then a divides c.
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If R is a unique factorization domain and if a,b are in R, then a and b have a least common multiple (l.c.m.) in R.
132136 If R is a unique factorization domain and if a,b are in R, then a and b have a least common multiple (l.c.m.) in R.
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Unique Factorization Domain with Quotient Field
This is a proof regarding a unique factorization domain.
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R-module proof
If R is a principal ideal domain.... Please see the attachment.
Proof:
Because is a principal ideal domain, then is also a unique factorization domain. For any nontrivial ideal in , we have is generated by some element .
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Simplification by Factorization and finding domain
182004 Simplification by Factorization and finding domain Problem 1) Simplify 5xy-15x/4xy
Problem 2) x2 -9x +18 = 0
(x-6)(x-3) = 0
x =6 x =3
What happens when x is 6 or 3. What is the domain if x can only be 6 or 3? 1.
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Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.
50267 Euclidean algorithm, primes and unique factorization, congruence Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial
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Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.
Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields
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the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.
Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields
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Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.
Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields