Algebra Qualifying Examination 6 January 2012 …

1 Algebra Qualifying Examination6 January 2012 instructions : There are 8 problems worth a total of 100 points. Individual point valuesare listed by each problem. Credit awarded for your answers will be based upon the correctness ofyour answers as well as the clarity and main steps of your must be written in a structured and understandable manner. You may use a calculator to check your computations, but it may not beused as a step in your reasoning. Every effort is made to ensure that there are no typographical errorsor omissions. If you suspect there is an error, check with the examadministrator. Do not interpret the problem in a way that makes :Throughout,QandCdenote the field of rational or complexnumbers, the ring of integers andFpdenotes the fieldwithpelements, wherepis a prime (11 points) LetKbe a field which is an extension of degreenof anotherfieldF, Kand [K:F] =n. Prove thatKis isomorphic to asubring of the ring ofn nmatrices overF. (Thus the ring ofn nmatrices overFcontains an isomorphic copy of every extension ofFofdegree n.)

2 2. (12 points) LetRbe a commutative ring with 1R, and letX={fi:i I}be a subset ofRsuch that the ideal generated byXis the unit ideal 1R = ) Show that a nitenumber of elements ofXgenerate the unit ) Sayf1, .. , fk Xgenerate the unit ideal. Show thatfn11, .. , fnkkgenerate the unit ideal forn1, .. , nkfixed positive ) Denote byRf the localization ofRat the multiplicative setS ={1R, f , f2 , f3 , ..}, =S 1 R, and let :R R be thecanonical homomorphism. Consider two elementsaanda ofRand suppose (a) = (a ) for 1 k. Showa=a . (We areassuming as in b) thatf1, .. , fkgenerate the unit ideal.)13. (14 points)a) LetGbe a simple group of ordernwith a proper subgroupHofindexk >1, [G:H] =k >1. Show thatGis isomorphic to asubgroup ofSk, the symmetric group ) SupposePis a Sylowp-subgroup ofGwhere we viewGas a sub-group ofSkas above. Show that ifPis also a Sylowp-subgroup ofSkthen the order of the normalizer ofPinG, |NG(P)|, divides|NSk(P)|.c) Use 3a and 3b above to show that a group of order 396 cannot besimple.

3 (Hint: Letp= 11 andH=NG(P).)4. (14 points)a) Recall that the ring of Gaussian integersZ[i] is a Euclidean domainand hence both a PID and a UFD. Use Eisenstein s criteria to showthatX4 3 is irreducible over the fieldQ(i).b) What is the Galois group of the splitting fieldFofX4 3 overQ(i) as an extension ofQ, givenQ Q(i) Ffind Gal(F/Q).Justify your ) Determine all the intermediate fieldsKwithQ K (10 points) LetRbe a ring with identity 1R(not necessarily commuta-tive). Show that the following conditions on a unitary (left)RmoduleMare )Mis injectiveii) every short exact sequence 0 M B C 0 (of unitary leftR-modules) is split exact, =M (13 points)a) Consider the symmetric groupSnwithn 3. SupposeNis anormal subgroup ofSnthat contains a 3-cycle. ShowNcontainsevery ) Show thatN= (13 points) LetVbe a two dimensional vector space over the fieldFpwithpelements (pan odd prime). LetL:V Vbe a lineartransformation onVsuch thatLp 1=I, the identity map. Prove thatLis diagonalizable, that is prove that there is a basisBforVsuch that[L]B, the matrix ofLwith respect toB, is a diagonal (13 points) LetRbe a commutative ring with 1R, and letI(Rbea proper ideal.

4 Show that there exists a minimal prime idealPoverI, a prime idealPsuch thatI Pand such that there does not existanother prime idealP withI P ()