{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# Algebra2006Jan - Department of Mathematics — Syracuse...

This preview shows pages 1–10. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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~by-n 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 3-cycles 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. ...
View Full Document

{[ snackBarMessage ]}

### Page1 / 10

Algebra2006Jan - Department of Mathematics — Syracuse...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online