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people:fer:504ws:spring2018:homework

Math 504 - Homework

  • LaTeX-ed solutions are encouraged and appreciated.
  • If you use LaTeX, hand-in a printed version of your homework.
  • You are encouraged to discuss homework problems with classmates, but such discussions should NOT include the exchange or written material.
  • Writing of homework problems should be done on an individual basis.
  • Outside references for material used in the solution of homework problems should be fully disclosed.
  • References to results from the textbook and/or class notes should also be included.
  • The following lists should be considered partial and tentative lists until the word complete appears next to it.
  • Use 8.5in x 11in paper with smooth borders. Write your name on top of each page. Staple all pages.

Problem Set 13 Due 05/07/2018 (complete)

  1. Let K and L be fields. Show that the set Hom(K,L) of all homomorphisms from K to L, is linearly independent over L. In particular Aut(K) is linearly independent over K.
  2. Prove that a finite group G is solvable iff there is a finite sequence of subgroups 1=H0H1Hn1Hn=G such that each Hi and H_{i+1}/H_i is cyclic. Show, with a counterexample, that this equivalence does not hold in general for arbitrary groups.
  3. Define: an angle \theta is constructible if there are two constructible straight lines forming an angle \theta.
    Prove: let l be a constructible straight line, A a constructible point on l, and \theta a constructible angle. The straight line(s) that go through A and form an angle \theta with l is(are) constructible.

Problem Set 12 Due 04/27/2018 (complete)

  1. Let F/K be a field extension, S\subseteq T\subseteq F with S algebraically independent over K, and F algebraic over K(T). Prove that there is a transcendence basis B, for F over K, such that S\subseteq B\subseteq T. (Hint: prove that a directed union of algebraically independent sets over K is algebraically independent over K, and use Zorn's lemma)
  2. Let F/K be a field extension and S\subseteq F. Prove that TFAE:
    1. S is maximal algebraically independent over K,
    2. S is algebraically independent over K and F is algebraic over K(S),
    3. S is minimal such that F is algebraic over K(S).
  3. Let F/E/K be a field tower. Prove that tr.d._K(F)=tr.d._E(F)+tr.d._K(E)
  4. Let K be a field, and t_1,\dots,t_n independent variables. If f(t_1,\dots,t_n)\in K[t_1,\dots,t_n] is a symmetric polynomial in variables t_1,\dots,t_n, there is a polynomial g, such that f(t_1,\dots,t_n) = g(s_1,\dots,s_n). (Hint: Use double induction on n and d, the total degree of f)

Old Homework

people/fer/504ws/spring2018/homework.txt · Last modified: 2020/01/10 14:43 by fer