Let R = Z[x] and let P = {f element of R | f(0) is an even integer}. Show that P is a prime ideal of R.
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Let R = Z[x] and let P = {f element of R | f(0) is an even integer}. Show that P is a prime ideal of R.
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- 34. If is an ideal of prove that the set is an ideal of . The set is called the annihilator of the ideal . Note the difference between and (of Exercise 24), where is the annihilator of an ideal and is the annihilator of an element of.Suppose 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.29. Let be the set of Gaussian integers . Let . a. Prove or disprove that is a substring of . b. Prove or disprove that is an ideal of .
- Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.Find a principal ideal (z) of such that each of the following products as defined in Exercise 10 is equal to (z). a. (2)(3)(4)(5)(4)(8)(a)(b)17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
- 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .Find the principal ideal (z) of Z such that each of the following sums as defined in Exercise 8 is equal to (z). (2)+(3) b. (4)+(6) c. (5)+(10) d. (a)+(b) If I1 and I2 are two ideals of the ring R, prove that the set I1+I2=x+yxI1,yI2 is an ideal of R that contains each of I1 and I2. The ideal I1+I2 is called the sum of ideals of I1 and I2.Prove that [ x ]={ a0+a1x+...+anxna0=2kfork }, the set of all polynomials in [ x ] with even constant term, is an ideal of [ x ]. Show that [ x ] is not a principal ideal; that is, show that there is no f(x)[ x ] such that [ x ]=(f(x))={ f(x)g(x)g(x)[ x ] }. Show that [ x ] is an ideal generated by two elements in [ x ] that is, [ x ]=(x,2)={ xf(x)+2g(x)f(x),g(x)[ x ] }.
