Consider the following polynomial over
Find each of the following polynomials with all coefficients in
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Chapter 8 Solutions
Elements Of Modern Algebra
- Write each of the following polynomials as a products of its leading coefficient and a finite number of monic irreducible polynomials over 5. State their zeros and the multiplicity of each zero. 2x3+1 3x3+2x2+x+2 3x3+x2+2x+4 2x3+4x2+3x+1 2x4+x3+3x+2 3x4+3x3+x+3 x4+x3+x2+2x+3 x4+x3+2x2+3x+2 x4+2x3+3x+4 x5+x4+3x3+2x2+4xarrow_forwarda. Find a nonconstant polynomial in Z4[ x ], if one exists, that is a unit. b. Find a nonconstant polynomial in Z3[ x ], if one exists, that is a unit. c. Prove or disprove that there exist nonconstant polynomials in Zp[ x ] that are units if p is prime.arrow_forwardSuppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive degree, and that f(x)=g(x)h(x). Prove that the zeros of f(x) in F consist of the zeros of g(x) in F together with the zeros of h(x) in F.arrow_forward
- 5. 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_forwardLabel each of the following statements as either true or false. We say that cF is a solution to the polynomial equation f(x)=0 if and only if f(c)=0inF.arrow_forward
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