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Unformatted text preview: Department of Mathematics — Syracuse University
Graduate Algebra Preliminary Examination January 27‘“, 2006 10 mi
10 Name: This exam contains 10 problems. You are not allowed to use notes or to
collaborate with one another. 1. Assume A is an n~byn matrix such that rk[A — 21n]— rk[(A — 21n)2] = 5. (Here rk denotes the rank of the matrix and In is the identity matrix.) How much can be concluded about the Jordan canonical form of A? 2. Let V = R2 be the Euclidean plane and assume that T: V —> V is a linear operator. Let l], 12 and 13
be three distinct lines passing through the origin with T(l,)= 1,» for i=1,2,3. Show that T is a dilation, that is, Tis multiplication by some constant. 3. Let k be a field and let T: V ~> W be a linear transformation between two vector spaces over k. (a) (2 points.) Define the transpose T. : W‘ —> V. of the linear operator T (b) (7 Points.) Show that T‘ is injective if and only if T is onto. 4. Show that a finite group of order 24 cannot be simple. 5. Let S" denote the symmetric group on n letters and let A" denote the alternating subgroup. Recall that if OEG, where G is a group, the centralizer of 0 in G is the subgroup
CG(O’)={‘L’EGITU=OT}. (a) (3 Points) If OE An, use the sign homomorphism from S" to {:1}, to show that CAH(O)’ the centralizer of or in An, is either equal to CS" (0) or it is a subgroup of CS"(0) of index 2. (b) (3 Points) If n = 5 and 0 E An is a 3—cycle, show that [Csn (a) : CA" (0)] = 2. (c) (4 Points) We know that all 3cycles are conjugate in S5. Use this and part (b) to show that all 3
cycles are conjugate in A5. 6. Let G be a finite p—group for some prime p. Show that the center of G is not trivial. 7. Let Q denote the ﬁeld of rational numbers and let f = X3 + 2X2 + 7 E Q[X]. (a) (3 Points.) Show that f has precisely one real root. (b) (3 Points.) Show that f is irreducible in Q[X]. (c) (4 Points.) Show the Galois group of f over Q is isomorphic to the symmetric group S3. 8. (a) (5 Points.) Let K/F be a ﬁnite extension of fields. Show that K/F is algebraic. (b) (5 Points.) Let UK and K/F be algebraic field extensions. Show that L/F is also algebraic. 9. (a) (3 Points) Assume R is a commutative ring and [Q R is an ideal. Show that I[X] Q R[X] is an ideal. (b) (7 Points.) Using the first isomorphism theorem or otherwise, show that‘ R[X]/I[X] is isomorphic to
(R/1)[X] 10. (a) (2 Points.) Let S be a commutative ring and assume that I: Sf + 5g is the ideal generated by two elements f and g. Show that if h E S is any element, then I is also generated by the elements
f and g— hf. (b) (4 Points.) Let'Z be the ring of integers and assume I is an ideal of Z[X] generated by the set
f,gEZ[X]. Show that we can replace f and g by two generators, one of which has a zero constant term. ...
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 Spring '11
 NA
 Algebra

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