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people:fer:504ws:spring2020:old_homework

Math 504 - Earlier Homework


Problem Set 6 Due 04/14/2020 (complete)

  1. Show that every φAutK(¯K) induces a complete lattice automorphism on SubK(¯K). (This is part of Prop. 4.6.3 in the posted class notes)
  2. Let E/K be an algebraic extension. Prove that the normal closure of E/K is the spliting field of the set of polynomials A={minK(α)αE×}. (This is Prop. 4.7.3 in the posted class notes)
  3. Let E/K be an infinite separable extension. Prove that [E:K]s=f[E:K], meaning both are finite and equal, or both are infinite. (Note that this and its converse were already proved for finite extensions as Prop. 3.71.3 in the posted class notes)
  4. Find an example of an algebraic extension for which [E:K]s=f[E:K], but E/K is NOT separable.

Problem Set 5 Due 03/24/2020 (complete)

  1. Let KEF, and αF, algebraic over K. Prove:
    1. If α is separable over K, then it is separable over E.
    2. If α is separable over E, and E/K is separable, then α is separable over K.
  2. Let K=F2(s,t) be the field of rational functions in two variables s and t, over the two element field, F2. Let α=s and β=t, i.e. α is a root of x2sK[x], and similarly for β. Prove or disprove that K(α,β) is a simple extension of K.
  3. Let K be a field of characteristic p.
    1. Show that K=K1/p iff K is perfect.
    2. Show that the field K1/p is a perfect field, and the smallest perfect field that contains K.
  4. Let K be a field of characteristic p. Is ¯K separable over K1/p? Prove or disprove.

Problem Set 4 Due 03/10/2020(complete)

  1. Show that the algebraic closure is a closure operator, i.e.
    1. K¯K,

    2. ¯¯K=¯K,
    3. KE¯K¯E.
  2. Let ¯K be an algebraic closure of K. Show:
    1. ¯K is minimal with the property of being an extension of K which is algebraically closed.
    2. ¯K is maximal with the property of being an algebraic extension of K.
  3. Let  f(x)K[x]. Prove that if α is a root of  f(x)  with multiplicity m, then α is a root of  f(i)(x)  for all  0i<m.
  4. Prove that if K is a perfect field, and F/K is an algebraic extension, then F is a perfect field.

Problem Set 3 Due 02/25/2020 (complete)

  1. Prove the corollary stated in class: If KEiF and each Ei/K is algebraic, then the join iIEi is algebraic over K.
  2. Let F/K be a finite extension. Prove that EndK(F)=AutK(F), i.e. every endomorphism of F that fixes K is an automorphism of F.
  3. Consider the extension F=Q(α,ω) of Q discussed in class, where α is a root of x32 and ω is a root of x2+x+1. Construct several automorphisms of F. Is there a bound for the number of automorphisms of F?
  4. Let F/K be a field extension, and φ:FL a field homomorphism. Let ˆF=φ(F) and ˆK=φ(K). Prove:
    1. [ˆF:ˆK]=[F:K].
    2. If F/K is algebraic, then so is ˆF/ˆK.
    3. If F/K is transcendental, then so is ˆF/ˆK.
    4. If F is an algebraic closure of K, then ˆF is an algebraic closure of ˆK.

Problem Set 02 Due 02/13/2020 (complete)

  1. Let F be a field extension of K, and a1,a2,,anF. Prove:
    1. K[a1,a2,,an]=K[a1][a2][an], and
    2. K(a1,a2,,an)=K(a1)(a2)(an).
  2. Show that Q( 2)Q( 3). Generalize.
  3. Grillet, Page 163, IV.2.1
  4. Grillet, Page 163, IV.2.2, IV.2.4

Problem Set 01 Due 02/04/2020 (complete)

  1. Let G be a group and NG. G is solvable iff N and G/N are solvable. In this case, l(G)l(N)+l(G/N).
  2. If L is a poset in which every subset has a l.u.b., then every subset of L also has a g.l.b.
  3. Given a lattice (L,,) in the algebraic sense, show that the binary relation , defined by xyiffxy=x, is a partial order on L. Moreover, for any x,yL, xy is the g.l.b.{x,y}, and xy is the l.u.b.{x,y}.
  4. Let A be a universal algebra, and Sub(A) the complete lattice of subuniverses of A. If DSub(A) is directed, then (XDX)Sub(A).

Homework

people/fer/504ws/spring2020/old_homework.txt · Last modified: 2020/04/23 23:12 by fer