QuestionQuestion

1. Inside the field C of complex numbers consider the subset of Gaussian integers Z[i] which, by definition, consists of all complex numbers a+bi where a and b are integers.
Check that Z[i] is an integral domain and identify its field of fractions (as a subset in C).
2. Let R be a non-zero ring, and let T be a subset of R containing neither 0 nor divisors of zero which is closed with respect to multiplication. Start with the set of pairs (a,b) with a from R and b from T, mimic the construction (see e.g. Section 21 of the textbook) of the field of fractions of an integral domain, to obtain a ring Q(R,T) containing R, and such that every element of T is invertible in Q(R,T). It is called the ring of partial fractions , a/b, with denominators in T.
(a) When R is an integral domain, what T one should take so that Q(R,T) becomes the field of fractions of R?
(b) What is Q(R,T) when R=Z (the ring of integers) and T is the subset consisting of all powers of 10?
(c) Let R be the ring R[x] of all polynomials with real coefficients in one indeterminate x. Let T be the subset of all those polynomials which have no real roots. Describe the ring Q(R,T) by identifying its elements with certain functions from R to R.
(d) In R=R[x], let T consist of of all powers (x-5)^n, n=0,1,2,3 ... of (x-5). Describe Q(R,T) by identifying its elements with certain functions on R-{5} (real numbers not equal to 5).
(e) In R=R[x], let T consists of all non-zero polynomials not divisible by (x-5). Describe Q(R,T) by identifying each of its elements with a certain functions in a neighborhood of x=5. Does there exist a neighborhood that would serve all elements of Q(R,T)?

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