MATH:5010 - Spring 2017

Reading (from Dummit and Foote):

Part A Problems (to be handed in on Tuesday, April 04, at the beginning of the discussion section):

• EXTRA problem on normal extensions: Let a be a real number such that a⁴ = 7. Let Q be the field of rational numbers, and let i be a square root of -1. Show that Q(ia²) is normal over Q. Show that Q(a+ia) is normal over Q(ia²). Show that Q(a+ia) is not normal over Q.
  HINT: To show that Q(a+ia) is not normal over Q, find the minimal polynomials of a+ia and a-ia over Q.

• Section 13.5: # 3, 4.
  IMPORTANT for #4: Fix an algebraic closure L of F_p. For all positive integers m, view F_p^m as the unique splitting field of x^pm-x inside L.

• Section 13.5: # 5.
  IMPORTANT: Since p can be greater than 3, it is not enough to show that the polynomial has no roots in F_p, but you have to also consider factors of degree > 1.

Part B Problems (to be handed in on Friday, April 07, at the beginning of the lecture):

• EXTRA problem on splitting fields: Let F be a field, let f(x) ∈ F[x] be of degree n ≥ 1, and let K be a splitting field of f(x) over F. Prove that [K:F] divides n!.
  HINT: Use induction on n and distinguish between the cases when f(x) is irreducible, resp. reducible, in F[x]. It may be helpful to remember that if n₁ + n₂ = n for positive integers n₁ and n₂, then the product n₁! n₂! divides n!.

• Section 13.5: # 7, 11.
  HINT for #7: Suppose a ∈ K has no p^th root in K. Show that f(x) = x^p - a is irreducible over K, but not separable. (Show that f(x) has a single root of multiplicity p. To show that f(x) is irreducible, you have to again consider possible factors of degree > 1, as in #5.)
  HINT for #11: You can use that we proved in class that if F is a perfect field then every irreducible polynomial in F[x] is separable.

Additional Practice Problems (you have to know how to do these, but you do NOT have to hand these in):

• Section 13.5: # 1, 2, 3, 4, 6, 8, 9.