Preliminary Algebra Examination, January 10, 2008
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Show your work, give reasons for your answers, provide all necessary proofs and
counterexamples. There are 10 problems on 18 pages worth 10 points each for the total of 100
points. Check that you have a complete exam, print your name on each page.
Problem 1
Problem 6
Problem 2
Problem 7
Problem 3
Problem 8
Problem 4
Problem 9
Problem 5
Problem 10
Total
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1. Let G, H be cyclic groups generated by elements x, y of nite orders m, n, respectively.
(a) (7 points) Determine the necessary and sucient condition on m, n so that sending xi to y i ,
for all i Z, is a well-dened homomoprhism of groups.
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1. (continued)
(b) (3 points) Describe all homomorphisms of the cyclic group of order 6 into the cyclic group
of order 24.
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2. Given a subgroup K of a group G, the set S of left cosets of K in G is a left G-set by means of
g xK = gxK , for all g, x G. If H is another subgroup of G, then S is a left H -set by restriction.
Recall that the set HxK = {y G | y = hxk for some h H, k K } is called a double coset. For
any set X , |X | denotes the cardinality of X .
(a) (3 points) Prove that the orbit of the element xK of the H -set S is the set of left cosets of
K in G contained in the double coset HxK and compute the stabilizer of xK .
(b) (2 points) Prove that the double cosets form a partition of G.
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2. (continued)
In the rest of the problem, assume |G| < .
(c) (3 points) Prove that |HxK | = |K |[H : H xKx1 ] = |H |[K : K x1 Hx].
(d) (2 points) Do all double cosets have the same cardinality? If yes, give a proof; if no, give a
counterexample.
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3. (10 points) Prove that if a group has order pe a where p is a prime, 1 a < p, and e 1,
then the group has a proper normal subgroup.
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4. Let A be a square matrix over the eld C of complex numbers.
(a) (3 points) Prove that the matrix is invertible if and only if all of its eigenvalues are dierent
from zero.
(b) (3 points) Prove that the matrix is nilpotent if and only if zero is its only eigenvalue. Recall
that a square matrix B is called nilpotent if B m = 0 for some positive integer m.
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4. (continued)
(c) (4 points) Prove that if A is nilpotent, it is similar to an upper triangular matrix with
diagonal entries zero. Recall that matrices X and Y are called similar if X = CY C 1 for some
invertible matrix C .
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5. Let A be a real symmetric n n matrix, and let T : Rn Rn be the linear operator on the
Euclidean space Rn given by T (X ) = AX , for all column vectors X Rn .
(a) (3 points) Prove that every vector in Ker T is orthogonal to every vector in Im T .
(b) (2 points) Prove that Rn = Ker T Im T .
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5. (5 points) Prove that T is an orthogonal projection onto Im T if and only if A, in addition
to being symmetric, satises A2 = A. Recall that for any subspace W Rn , the equality Rn =
W W says that every vector v Rn can be uniquely written as v = w + w , where w W and
w W . The linear operator on Rn sending v to w, for all v , is called the orthogonal projection
onto W .
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6. (10 points) Let Z[x] be the ring of polynomials in one variable with coecients in the integers.
Let (3, x) = M Z[x] be the ideal generated by 3 and x. Prove that M is a maximal ideal.
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7. Let R be a commutative ring with identity. Let I and J be ideals of R. Recall that IJ equals
the ideal generated by {ij | i I, j J }.
(a) (4 points) Prove that IJ I J .
(b) (3 points) Give an example where IJ = I J . Make the example nontrivial in the sense
that neither I nor J equals either 0 or R.
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7. (continued)
(c) (3 points) Give an example where IJ = I J .
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8. Let F be a eld and F [x] the ring of polynomials in one variable with coecients in F .
(a) (2 points) Show that a module M over F [x] is also in a natural way a vector space over F .
(b) (4 points) Assume that F is algebraically closed and that M is a simple module over F [x].
Prove that the dimension of M as a vector space over F is one.
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8. (continued)
(c) (4 points) Assume that F is not algebraically closed. Prove that there exists a simple module
M over F [x] such that the dimension of M as a vector space over F is greater than one.
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9. Let T be a linear operator on a nite dimensional vector space over the complex numbers.
Assume that T has two eigenvalues: 3,4. Assume that the Jordan canonical form of a matrix
representing T has the following form. For the eigenvalue 3 there are 2 blocks of size 1, 2 blocks
of size 2, and 1 block of size 4. For the eigenvalue 4 there are 1 block of size 1, 3 blocks of size 3,
and 1 block of size 5.
(a) (4 points) What is the characteristic polynomial of T ?
(b) (3 points) What is the minimal polynomial of T ?
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9. (continued)
(c) (3 points) What is the nullity of (T 3I )3 ? I is the identity linear transformation.
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10. (10 points) Let F K be an extension of elds of characteristic 0. Let G be the Galois
group of K over F . We do not assume that the eld extension F K is a Galois extension.
Assume that G is a nite group and that p is a prime number that divides the order of G. Prove
that there exists a eld L with F L K satisfying all of the following properties.
(a) L K is a Galois eld extension with Galois group isomorphic to Z/pZ.
(b) The degree of the eld extension L K is p.
(c) There does not exist any eld strictly between L and K .
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