B. Problems Involving Concepts and Definitions 1 Is x8 + 1 =x3 + 1 in zslx)? Exp
ID: 3184095 • Letter: B
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B. Problems Involving Concepts and Definitions 1 Is x8 + 1 =x3 + 1 in zslx)? Explain your answer 2 Is there Explain. # 3 Write all the quadratic polynomials in zs[x] How many are there? How many cubic polynomials ar any ring A such that in Alx], some polynomial of degree 2 is equal to a polynomial of degree 4? there in zslx]? More generally, how many polynomials of degree m are there in znx]? 4 Let A be an integral domain; prove the following: If(x + 1 )2 = x2 + 1 in A[x], then A must have characteristic 2. If ( +144+1 in A[x], then A must have characteristic 2. If (r16+2+ in Ax], then A must have characteristic 3. 5 Find an example of each of the following in zslx!: a divisor of zero, an invertible element. (Find nonconstant examples.) 6 Explain why r cannot be invertible in any Alx], hence no domain of polynomials can ever be a field. 7 There are rings such as P3 in which every element 0,l is a divisor of zero. Explain why this cannot happen in any ring of polynomials A[x], even when A is not an integral domain. 8 Show that in every Alr], there are elements #0,1 which are not idempotent, and elements #0,1 which are not nilpotent.Explanation / Answer
2.No. In the ring of polynomials two polynomials are defined to be equal iff their coefficients agree, and degree of a polynomial is the largest k for which the coefficient of x^k is nonzero.
5.Zero divisor examples: 2x,4x,6x
an invertible element :(1+4x)(1+4x).
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