Algebra Qualifying Examination, September 2002
Answer all the problems and show all your works.
1. (15%) Let G be a nonabelian group of order 6. Show that G is isomorphic to S3, the symmetry group of degree 3.
2. (15%) Let G be a group of order 56. Suppose that G has no element of order 14.
Show that the Sylow 2-subgroup of G is normal in G.
∈
→≤
∈ ≠
σ σ σ
3. (20%) Let G is a group of order 231. Show that the Sylow 11-subgroup of G is in the center of G.
4. (15%) Let R be a commutative ring with identity and f(x) = a0+a1x+…..+anxn ∈R{x}.
Show that f(x) is a unit in R[x] if and only if a0 is a unit in R and a1,…..,an are nilpotent elements in R.
5. (10%)Let R be a integral domain and a, b R. Suppose an= bn and am = bm, where m,n are positive integers and (m,n) = 1. Prove that a = b.
6. (10%)An integral domain D is called a Euclidean domain if there is a function d: D\{0} Z+ such that
(1). d(a) d(ab) for any a, b∈D\{0} and
(2). for any a D and b 0, there are q,r∈D such that a = qb + r, where d(r) < d(b) or r = 0.
Show that d(a) == d(e) if and only if a is a unit.
7. (15%) (i) Show that a finite extension E of F is also an algebraic extension of F.
(ii) Let K be a field and E an extension of K. Suppose u, v∈E are roots of an irreducible polynomial f(x) £∈ K[x]. Show that there is a unique field isomorphism
: K(u)- K(v) such that |K = idK and (u) = v.
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