Rings and Fields — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

3. Attempt two parts from each question. 1. (a) Let R be the ring of all real-valued continuous functions defined on the closed unit interval. Let M = {f € R | f(y) = 0), where y E [0, 1]. Show that M is a maximal ideal of R. (b) Let R be a Euclidean ring, and let a, b e R. If b ‡ 0 is not a unit in R, then d(a) < d(ab). (c) Show that the ideal generated by a prime element in a commutative ring with unity is a prime ideal. 2. (a) Let D be a principal ideal domain and let p e D. Prove that (p) is a maximal ideal in D if and only if p is irreducible. (b) In a Euclidean ring, prove that any two greatest common divisors of a and b are associates. (c) Let R be a Euclidean ring. Then any two elements a, b E R have a greatest common divisor d. Moreover, d= La + ub for some 2, u E R. 3. (a) If pis a prime number of the form 4n+1, then prove that p=2 + b for some integers a and b. (b) Let F[x] be the field of polynomial rings and if f(x) and g(x) are two nonzero elements of F[x], then deg(f(x)g(x)) = deg(f(x))) + deg(g(x)). 4. (c) Prove that x2+¥+4 is irreducible over F, the field of integers mod 11 and show that #isafieldhavingl21elements. (a) a) Let p is a prime number, then prove that the pc>lynomial xn.-p is irreducible over the rationals. . ql) The polynomial rf +4 is irreducible or reducible over the field of rationals. Justify your answer. (b) a) Prove that a principal ideal ring is a unidue factorization domain. aD Prove that giv=§]={a+b\/=3| a, d E4 be the ring of Gaussian integers is not a unique factorization domain. (a) Let Fbe a field and i(x) a non-constant polynomial in fl.x']. Then prove that there is an extension field Eof Fin which ¢Jr) has a zero. Also find the splitting field of ;r4+1 5. (a) (i} IfE is a finite extension ofF then E is an algebraic extensic)n ofF. (ii) Find the value of [GF(729) : GF(9)]. (b) (i) Draw a subfie]d lattice of GF(3]8) and GF(230). (ii) Define constructible number. (c) Let x be a zero of f{x) = x2+ x + 1 in®some extension of z2, Find the other zero of f{x) in Z2[x]. (a) If a and b are constructible numbers give a geometric proof that a+b and a-b are constructible. (c) Show that Q[V2 , V3] = Q[V2 + ly3].