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Algebra2005Jan - Ph D Algebra Preliminary Exam Show the...

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Unformatted text preview: Ph. D. Algebra Preliminary Exam January 29, 2005 Show the work you do to obtain an answer. Give reasons for your answers. 1. Consider the following set of vectors in R3 1345 S={2,5,7,9}. 1345 Find a subset T C S such that T is a basis for the span of S. 2. Let T be the linear transformation from R2 to R2 defined by T(:1:, y) = (a: + y,x — y). Determine all ordered bases B for R2 such that the matrix representing T with respect to B (the same B being used as the ordered basis for both the domain R2 and the target R2) equals [‘3 i]- To help make it clear that you really understand your description of all such B, do the following. State explicitly Whether the number of such B is 0, 1, a finite number greater than 1, or infinite. If the number is 1, 2, or 3, list them explicitly. If the number is greater than 3, list at least 3 different answers explicitly. If your description of all such B is a good one doing those explicit things should be a triviality. 3. A square matrix A has characteristic polynomial (x — 1)6(:z: — 2)4, nullity (A — [)23, nullity (A — I)2=5, nullity(A — 2])22, and nullity(A — 202:4. What is the Jordan normal form for A? 4. Let V be an inner product space with inner product ( , ) and u and v vectors in V. Prove that u = v if and only if (u, w) = (71,10) for all M) E V. 5. For this one you do not have to show work. We are just testing to see if you remember a famous theorem. Fill in the blanks to complete the following famous theorem. Theorem. If A is a given m X n matrix then, (a) The null space of A is the orthogonal complement of (blank). (b) The null space of AT is the orthogonal complement of (blank). 1 6. Let g, h be elements of a group G. If 9% = hg4 and g7 = 1, prove that gh = hg. 7. If H is a subgroup of a group G, then G acts on the set G/H of left cosets of H in G by g - xH = gasH. Describe the stabilizer of the coset aH explicitly as a subgroup of G. 8. (a) Prove that an integral domain with finitely many elements is a field. (b) Is there an integral domain containing exactly 10 elements? 9. For a prime p, the cyclotomic polynomial xp‘l + mp4 + - -~ + :r + 1 is irreducible in QM. Use this fact to prove the following statement. If C = 627”” and n = 6277175 then 7] ¢ Q((). 10. Let K be a splitting field of an irreducible cubic polynomial f (9:) over a field F of characteristic 0 Whose Galois group is 53. If a E K satisfies f (oz) = 0, determine the group of automorphisms C(F (a) / F) of the extension F (a). ...
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