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people:fer:402ws:spring2019:homework

Math 402 - 01 Homework (Spring 2019)


Problem Set 10 (complete) Due: 05/10/2019

  1. Let F be a field of characteristic zero, aF, and ξ=ξn a primitive n-th root of unity.
    1. Show by example that GalF(F(ξ)) need not be all of Un.
    2. Show by example that GalF(ξ)(F(ξ,na)) need not be all of Cn.
  2. Let G and H be solvable groups. Prove that G×H is solvable.
  3. Show that the change of variable y=x+(a/3) transforms the general cubic equation x3+ax2+bx+c=0 into a depressed cubic. Therefore, Cardano's formula is useful to solve any cubic equation.

Problem Set 09 (complete) Due: 05/03/2019 Board presentation: 05/10/2019

  1. Prove that the homomorphism ψ:UnGal(Q(ξn)/Q)kψk is surjective and injective.
  2. Let ξ15=cis(2π/15) be a primitive 15-th root of unity.
    1. Find the group Gal(Q(ξ15)/Q) and draw its lattice of subgroups.
    2. Find and draw the lattice of intermediate fields of the extension Q(ξ15)/Q.
    3. Write down the correspondence between the subgroups in part 1, and the subfields in part 2, using the Fundamental Theorem of Galois Theory.
  3. Show that any non-abelian simple group is non-solvable.
  4. Show that if d is a divisor of n then Q(ξd) is a subfield of Q(ξn). Conclude that φ(d) divides φ(n), and Ud is a quotient of Un.

Problem Set 08 (complete) Due: 04/26/2019 Board presentation: 05/03/2019

  1. Prove the following corollary to the Fundamental Theorem of Galois Theory. Use only the FTGT statements to prove it. Let E/F be a (finite) Galois extension, with Galois group G=GalF(E). Let L1,L2SubF(E) and H1,H2Sub(G).
    1. (L1L2)=L1L2
    2. (L1L2)=L1L2
    3. (H1H2)=H1H2
    4. (H1H2)=H1H2
  2. Let f(x)Q[x] be such that it has a non-real root. Let E be the splitting field of f(x) over Q. Prove that GalQ(E) has even order.
  3. Consider the polynomial f(x)=x3+2x2+2x+2Q[x], and E its splitting field over Q.
    1. Show that f(x) has exactly one real root. (Hint: use calculus)
    2. Show that f(x) is irreducible over Q.
    3. Find [E:Q]. Fully explain your calculation.
    4. Determine GalQ(E).
  4. Consider the group Sn of all permutations of the set {1,2,,n}.
    1. Show that the transpositions (1  2),(2  3),,(n1  n) generate the whole group Sn.
    2. Show that Sn is generated by the following two permutations: ρ=(1  2    n)andσ=(1  2) (Hint: conjugate σ by ρ.)
    3. For p is a prime, ρ a p-cycle, and σ a transposition, show that ρ and σ generate Sp. Show, by counterexample, that the hypothesis of p being prime cannot be removed.

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people/fer/402ws/spring2019/homework.txt · Last modified: 2019/05/09 09:11 by fer