Abstract Algebra Prelim Aug. 2009
1. For a∈Z/nZ, let πa:Z/nZ→Z/nZby πa(x) = x+a. This is a permutation on Z/nZ.
(a) Compute the order of πaas a permutation. Your answer will depend on nand a.
(b) Determine when πais an even permutation. Your answer will depend on nand a.
2. State and prove the Eisenstein criterion for polynomials in Z[X].
3. Let Rbe a commutative ring and define Jto be the intersection of all maximal ideals of R.
(a) Prove that Jis an ideal.
(b) Let a∈J. Prove that for all b∈R, the element 1 −ab is invertible.
4. Let F1, . . . , Fnbe fields, where n≥2, and set A=F1× · · · × Fn(the product ring). For any
subset Sof {1, . . . , n}, let
IS={(x1, . . . , xn)∈A:xi= 0 for i∈S},
so in particular I∅=Aand I{1,...,n}={0}. (Smaller Smake larger IS.)
(a) Prove ISis an ideal in Aand describe the ring A/ISin terms of a product of fields.
(b) For any x= (x1, . . . , xn)∈A, describe the principal ideal Axin terms of the coordinates
of x.
(c) Show every ideal in Ahas the form ISfor some subset Sof {1,. . . , n}.
5. Let Gbe a finite group which acts on a set X.
(a) Let N={g∈G:gx =xfor all x∈X}. Show Nis the largest normal subgroup of G
contained in the stabilizer subgroup of each point in X.
(b) If the action has one orbit, show for any two points xand yin Xthat their stabilizer
subgroups are conjugate.
(c) Let G=S3be the permutation group on 3 elements. Give two examples of actions of G
on itself where
(i) there is only one orbit,
(ii) there is more than one orbit and the conclusion of part (b) is false.
Provide a brief explanation of why your answers to (i) and (ii) fit the conditions.
6. Give examples as requested, with brief justification.
(a) A nonabelian group of order 27.
(b) A prime element of Z[i].
(c) A cyclic group with exactly 8 generators.
(d) A free module (over some ring), and a nonzero submodule which is not free.