Some known results on polynomial factorization over towers of field
... We consider the polynomial ring S[t1 , . . . , tn ], with either: • S=Z • or S = Fq , with q a prime power, and in this case n > 0. We let K be the fraction field of S and introduce the field of fractions K(t1 , . . . , tn ); we are interested in a field extension L of K(t1 , . . . , tn ) of the for ...
... We consider the polynomial ring S[t1 , . . . , tn ], with either: • S=Z • or S = Fq , with q a prime power, and in this case n > 0. We let K be the fraction field of S and introduce the field of fractions K(t1 , . . . , tn ); we are interested in a field extension L of K(t1 , . . . , tn ) of the for ...
10.3 Simplified Form for Radicals
... If bn = a, then b is an nth root of a and b = a1/n = a , where a is called the radicand and n is called the root index. If n is a positive even integer and a is a positive real number, then a1/n is the positive real nth root of a and is called the principal root. If n is a positive odd integer and a ...
... If bn = a, then b is an nth root of a and b = a1/n = a , where a is called the radicand and n is called the root index. If n is a positive even integer and a is a positive real number, then a1/n is the positive real nth root of a and is called the principal root. If n is a positive odd integer and a ...
MA314 (Part 2) 2012-2013 - School of Mathematics, Statistics
... These definitions make no reference to any particular set or to the nature of the objects that are elements of the ring - whether they are numbers, functions, matrices or whatever. When thinking about the definitions, try to bear this in mind and try also to bear in mind that “addition” and “multipl ...
... These definitions make no reference to any particular set or to the nature of the objects that are elements of the ring - whether they are numbers, functions, matrices or whatever. When thinking about the definitions, try to bear this in mind and try also to bear in mind that “addition” and “multipl ...
MA554 Workshop 3
... 4. Let f = t3 + 2t2 + 3t + 4 and g = t2 + 1. (These are elements of the ring R[t], for example.) By using long division of polynomials, find polynomials q, r so that f = qg + r and r = 0 or deg r < deg g. Repeat this exercise with f = t4 − 2t3 + 3t − 2 and g = t − 1. Repeat this exercise with f = t4 ...
... 4. Let f = t3 + 2t2 + 3t + 4 and g = t2 + 1. (These are elements of the ring R[t], for example.) By using long division of polynomials, find polynomials q, r so that f = qg + r and r = 0 or deg r < deg g. Repeat this exercise with f = t4 − 2t3 + 3t − 2 and g = t − 1. Repeat this exercise with f = t4 ...
6.6. Unique Factorization Domains
... We say that a1 , . . . , as are relatively prime if 1 is a greatest common divisor of {a1 , . . . , as }, that is, if a1 , . . . , as have no common irreducible factors. Remark 6.6.3. In a principal ideal domain R, a greatest common divisor of two elements a and b is always an element of the ideal a ...
... We say that a1 , . . . , as are relatively prime if 1 is a greatest common divisor of {a1 , . . . , as }, that is, if a1 , . . . , as have no common irreducible factors. Remark 6.6.3. In a principal ideal domain R, a greatest common divisor of two elements a and b is always an element of the ideal a ...
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Principal Ideal Domains
... This section of notes roughly follows Sections 8.1-8.2 in Dummit and Foote. Throughout this whole section, we assume that R is a commutative ring. Definition 55. Let R b a commutative ring and let a, b ∈ R with b , 0. (1) a is said to be multiple of b if there exists an element x ∈ R with a = bx. In ...
... This section of notes roughly follows Sections 8.1-8.2 in Dummit and Foote. Throughout this whole section, we assume that R is a commutative ring. Definition 55. Let R b a commutative ring and let a, b ∈ R with b , 0. (1) a is said to be multiple of b if there exists an element x ∈ R with a = bx. In ...
Math 5c Problems
... 6. Let m; n 2 Z be square free integers and let = m + n . Let m be its minimal polynomial over Q p a) show that deg m =4 if and only if mn 2 Z. b) let p 2 Z be a prime and ': Z ! F p the mod p map. Show at least one of x2 ¡ '(n), x2 ¡ '(m); x2 ¡ '(nm) is reducible in F p[x] c) Assume that deg m ...
... 6. Let m; n 2 Z be square free integers and let = m + n . Let m be its minimal polynomial over Q p a) show that deg m =4 if and only if mn 2 Z. b) let p 2 Z be a prime and ': Z ! F p the mod p map. Show at least one of x2 ¡ '(n), x2 ¡ '(m); x2 ¡ '(nm) is reducible in F p[x] c) Assume that deg m ...