Math 402 - Spring 2016

Old Homework

Homework Problems

Problems to hand in

Problems to present in class
Problem Set 10, Friday 04/22/2016 (complete)
  1. Page 420, 10.1.6
  2. Page 420, 10.1.10,13
  3. Show that over a field of characteristic p, a polynomial of degree p-1 does not have an antiderivative.
  4. Let p be a prime, and F=ℤp(t) the field of rational functions over F=ℤp. Show that the polynomial xp-t is irreducible over F, and it has a root of multiplicity p.
Some additional problems. Will not to be collected.
  • Page 421, 10.1.18
  • Let f ∈ F[x], and a,b ∈ F, with a ≠ 0. Show that f(x) is irreducible over F iff f(ax+b) is irreducible over F.
Wednesday 04/27/2016 (complete)
  1. Page 420, 10.1.6
  2. Page 420, 10.1.10,13
  3. Show that over a field of characteristic p, a polynomial of degree p-1 does not have an antiderivative.
  4. Let p be a prime, and F=ℤp(t) the field of rational functions over F=ℤp. Show that the polynomial xp-t is irreducible over F, and it has a root of multiplicity p.
  5. Page 421, 10.1.18
  6. Let f ∈ F[x], and a,b ∈ F, with a ≠ 0. Show that f(x) is irreducible over F iff f(ax+b) is irreducible over F.
Problem Set 9, Monday 04/11/2016 (complete)
  1. Let F be a field that has no proper finite extension. Show that F is algebraically closed.
  2. Page 303, 6.4.4.b
  3. Page 303, 6.4.6
  4. Page 304, 6.4.20 (Hint: use Cor 2, page 303)
Some additional problems. Will not to be collected.
  • Page 304, 6.4.22
  • How many monic irreducible polynomials of degree 10 are there in ℤ2[x]?
  • Show that the polynomial f=x5 + 2x + 4 is irreducible over ℚ, and it has exactly one real root. Show that f has roots α1 and α2 such that ℚ(α1) ≈ ℚ(α2), but ℚ(α1) ≠ ℚ(α2).
Thursday 04/14/2016 (complete)
  1. Let F be a field that has no proper finite extension. Show that F is algebraically closed.
  2. Page 303, 6.4.4.b
  3. Page 303, 6.4.6
  4. Page 304, 6.4.20 (Hint: use Cor 2, page 303)
  5. Page 304, 6.4.22
  6. How many monic irreducible polynomials of degree 10 are there in ℤ2[x]?
  7. Show that the polynomial f=x5 + 2x + 4 is irreducible over ℚ, and it has exactly one real root. Show that f has roots α1 and α2 such that ℚ(α1) ≈ ℚ(α2), but ℚ(α1) ≠ ℚ(α2).
Problem Set 8, Monday 04/04/2016 (complete)
  1. Page 290, 6.2.29,30
  2. Page 297, 6.3.11
  3. Find the minimal polynomial minℚ(u) where u = (∛  2  + ω), and ω = cis(2π/3).
  4. Finish the computation of Aut(ℚ(∛  2 ,ω)) began in class. Which group is this? Why?
Some additional problems. Will not to be collected.
  • Page 291, 6.2.33
  • Page 297, 6.3.9
  • Page 297, 6.3.13
Friday 04/08/2016 (complete)
  1. Page 290, 6.2.29,30
  2. Page 297, 6.3.11
  3. Find the minimal polynomial minℚ(u) where u = (∛  2  + ω), and ω = cis(2π/3).
  4. Finish the computation of Aut(ℚ(∛  2 ,ω)) began in class. Which group is this? Why?
  5. Page 291, 6.2.33
  6. Page 297, 6.3.9
  7. Page 297, 6.3.13
Problem Set 7, Friday 03/18/2016 (complete)
  1. Page 289, 6.2.3.b,d (Hint: use the fact that π is transcendental over ℚ)
  2. Page 290, 6.2.7.b
  3. Page 290, 6.2.20
  4. Page 291, 6.2.32
Some additional problems. Will not to be collected.
  • Page 290, 6.2.21
  • Page 290, 6.2.27
Wednesday 03/23/2016 (complete)
  1. Page 289, 6.2.3.b,d (Hint: use the fact that π is transcendental over ℚ)
  2. Page 290, 6.2.7.b
  3. Page 290, 6.2.20
  4. Page 291, 6.2.32
  5. Page 290, 6.2.21
  6. Page 290, 6.2.27
Problem Set 6, Friday 03/11/2016 (complete)
  1. Page 225, 4.2.18.b,d,f
  2. Page 226, 4.2.21.b,d, 4.2.22.b
  3. Page 264, 5.1.37
  4. Page 281, 6.1.9
Some additional problems. Will not to be collected.
  • Page 263, 5.1.28
  • Page 282, 6.1.24
  • Page 282, 6.1.25
Friday 03/18/2016 (complete)
  1. Page 225, 4.2.18.b,d,f
  2. Page 226, 4.2.21.b,d, 4.2.22.b
  3. Page 264, 5.1.37
  4. Page 281, 6.1.9
  5. Page 263, 5.1.28
  6. Page 282, 6.1.24
  7. Page 282, 6.1.25
Problem Set 5, Friday 02/26/2016 (complete)
  1. Show that the Gaussian integers, ℤ(i) form an Euclidean Domain. Hint: Use division in the field ℂ of complex numbers to get a first approximation to the quotient.
  2. Page 225, 4.2.16
  3. Page 263, 5.1.16
  4. Page 272, 5.2.19
Some additional problems. Will not to be collected.
  • Page 263, 5.1.14
  • Page 263, 5.1.23
  • Page 272, 5.2.24
Friday 03/04/2016 (complete)
  1. Show that the Gaussian integers, ℤ(i) form an Euclidean Domain. Hint: Use division in the field ℂ of complex numbers to get a first approximation to the quotient.
  2. Page 225, 4.2.16
  3. Page 263, 5.1.16
  4. Page 272, 5.2.19
  5. Page 263, 5.1.14
  6. Page 263, 5.1.23
  7. Page 272, 5.2.24
 
Problem Set 4, Friday 02/19/2016 (complete)
  1. Page 212, problem 4.1.14.b,d,f
  2. Page 213, problem 4.1.23.b,d
  3. Page 225, problem 4.2.4.b,d,f
  4. Page 225, problem 4.2.10
Some additional problems. Will not to be collected.
  • Page 212, problem 4.1.11
  • Page 213, problem 4.1.31
Wednesday 02/24/2016 (complete)
  1. Page 212, problem 4.1.14.b,d,f
  2. Page 213, problem 4.1.23.b,d
  3. Page 225, problem 4.2.4.b,d,f
  4. Page 225, problem 4.2.10
  5. Page 212, problem 4.1.11
  6. Page 213, problem 4.1.31
 
Problem Set 3, Friday 02/12/2016 (complete)
  1. Page 198, problems 3.4.32 and 33
  2. Page 212, problem 4.1.3
  3. Page 212, problem 4.1.6
  4. Page 213, problem 4.1.21
Friday 02/19/2016 (complete)
  1. Page 198, problems 3.4.32 and 33
  2. Page 212, problem 4.1.3
  3. Page 212, problem 4.1.6
  4. Page 213, problem 4.1.21
 
Problem Set 2, Friday 02/05/2016 (complete)
  1. Page 178, problem 3.2.21
  2. Page 178, problem 3.2.26
  3. Page 186, problem 3.3.3
  4. Pate 187, problem 3.3.20
Wednesday 02/10/2016 (complete)
  1. Page 178, problem 3.2.21
  2. Page 178, problem 3.2.26
  3. Page 186, problem 3.3.3
  4. Pate 187, problem 3.3.20
Problem Set 1, Monday 02/01/2016 (complete)
  1. Page 169, problem 3.1.3
  2. Let R be a commutative ring of characteristic p (a prime). Show that the map ϕ:R → R given by x ↦ xp is a ring endomorphism.
  3. Page 169, problem 3.1.13
  4. Page 170, problems 3.1.33 and 34
Wednesday 02/03/2016 (complete)
  1. Page 169, problem 3.1.3
  2. Let R be a commutative ring of characteristic p (a prime). Show that the map ϕ:R → R given by x ↦ xp is a ring endomorphism.
  3. Page 169, problem 3.1.13
  4. Page 170, problems 3.1.33 and 34