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1. Problem Which of the following are groups under addition, multiplication, Rings and fields? If not, why not (explicitly state the properties they do not satisfy) Group Group under Ring with Field under multiplication identity addition (After removing the additive identity) Natural Numbers (1,2,3, Under ordinary addition and multiplication Whole Numbers (0,1,2,3,.. ) Under ordinary addition and multiplication Integers Under ordinary addition and nultiplication Rational Numbers Q Under ordinary addition and multiplication Real Numbers Under ordinary addition and aultiplication Complex Numbers Under complex number addition and multiplication Z. Under Mod addition and multiplication Z2 Under Mod addition and multiplication 2. Problem Which of the following polynomials defined over the field of real numbers are irreducible? f(x) x+1 g(x) =x-1 h(x=x2+1 p(x)=x2-1 q(x) + : r(x) 3. Problem Multiply the following polynomials defined over set of real numbers: f(x) x2 and g(x) =x+ 1. 4. Problem Using polynomial long division, divide f(x) by g(x) defined over the set of real numbers: f(x) x+ x2 and g(x) x + 1. Find g(x) andr(x) such that f(x) = q(x)g(x) r(x) 5. Problem Which of the following polynomials defined over the field of GF(2) are irreducible? f(x) 1 g(x) =x-1 h(x) =x2+1 p(x) x2-1 q(x) =x++1 r(x) =x³ 6. Problem Multiply the following polynomials defined over GF(2): f(x) x+ +x2+ and g(x) = = x +1. 7. Problem Using polynomial long division. divide f(x) by g(x) defined over GF(2) f(x) = x4 x2+ and g(x) + 1. Find q(x) and r(x) such that f(x) = q(x)g(x) r(x) 8. Problem In this problem all polynomials are over GF(2): f(x) X2 + and g(x) =x + 1 Using extended Euclidean algorithm find s(x) and t(x) such that: god(f(x),g(x)) s(x)f(x) t(x)g(x) Hint: the procedure is identical to one we used for two numbers a and b where we discovered and y Here integers a and are replaced by polynomials f(x) and g(x) and the linear coefficients : and are replaced by polynomials that you have to find s(x) and t(x). Use long division.

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Groups and Fields Problems
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