Write each of the following polynomials as a products of its leading coefficient and a finite number of monic irreducible polynomials over
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Elements Of Modern Algebra
- Consider the following polynomial over Z9, where a is written for [ a ] in Z9: f(x)=2x3+7x+4, g(x)=4x2+4x+6, h(x)=6x2+3. Find each of the following polynomials with all coefficients in Z9. a. f(x)+g(x) b. g(x)+h(x) c. f(x)g(x) d. g(x)h(x) e. f(x)g(x)+h(x) f. f(x)+g(x)h(x) g. f(x)g(x)+f(x)h(x) h. f(x)h(x)+g(x)h(x)arrow_forward5. Decide whether each of the following subset is a subring of , and justify your decision in each case. a. The set of all polynomials with zero constant term. b. The set of all polynomials that have zero coefficients for all even powers of . c. The set of all polynomials that have zero coefficients for all odd powers of . d. The set consisting of the zero polynomials together with all polynomials that have degree 2 or less.arrow_forwardLet Q denote the field of rational numbers, R the field of real numbers, and C the field of complex. Determine whether each of the following polynomials is irreducible over each of the indicated fields, and state all the zeroes in each of the fields. a. x22 over Q, R, and C b. x2+1 over Q, R, and C c. x2+x2 over Q, R, and C d. x2+2x+2 over Q, R, and C e. x2+x+2 over Z3, Z5, and Z7 f. x2+2x+2 over Z3, Z5, and Z7 g. x3x2+2x+2 over Z3, Z5, and Z7 h. x4+2x2+1 over Z3, Z5, and Z7arrow_forward
- Find the quartic (fourth-degree) polynomial function f with real coefficients that has 1,2and2i, as zeros, and f(1)=10.arrow_forwardLet [ a ] be an element of n that has a multiplicative inverse [ a ]1 in n. Prove that [ x ]=[ a ]1[ b ] is the unique solution in n to the equation [ a ][ x ]=[ b ].arrow_forward1. Find a monic polynomial of least degree over that has the given numbers as zeros, and a monic polynomial of least degree with real coefficients that has the given numbers as zeros. a. b. c. d. e. f. g. and h. andarrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning