5.9. (a) Let K/F be an extension of fields. Prove that [K: F] = 1 if and only if K= F. (b) Let L/F be a finite extension of fields, and suppose that [L: F] is prime. Suppose further that K is a field lying between F and L; i.e., F C K C L. Prove that either K = F or K = L.

Theorem 1.2.17 (Intervals) In an ordered field F, the following sets are intervals: (a) [a, b] = {x E F:a ≤x≤b}; (This could be {a} or Ø.) (b) (a, b) = {x E F:a < x

Exercise 4.2. (Bezout's identity for polynomials) Let F be a field. Let f, g E F[x] with greatest common divisor d. Prove that there exist polynomials u and v such that fu+gv = d.

True or False1. Every nonconstant polynomial in F[x] has a root in every extension field of F, that is if K is an extension of F and f is a nonconstant polynomial in F[x], then f has a root in K.2. There is no proper intermediate field of ℝ and ℂ, i.e. there exists no field E such that ℝ ≤ E ≤ ℂ but ℝ ≠ E, E ≠ ℂ.3. If [K:F]=p for some prime p and F ≤ E ≤ F, then F=E.