Higher Algebra 2 — Spring 2006

Gerald Hoehn

Problem sheet 4 March 3, 2006

Problem 1: Let k be a field of characteristic different from 2.

(a) Show that K/k is an extension of degree 2 if and only if K of the form k(\sqrt{D}) where D is an element of k which is not a square in k. Here, \sqrt{D} denotes a root of the polynomial x^{2} − D.

(b) Let D_{1} and D_{2} be elements of k, neither of which is a square in k. Prove that k(\sqrt{D_{1}}, \sqrt{D_{2}}) is of degree 4 over k if D_{1}D_{2} is not a square in k and is of degree 2 otherwise.

Problem 3: Determine the splitting fields and its degrees over Q of the following polynomials:

(a) x^{4} − 2,

(b) x^{4} + 2,

(c) x^{4} + x^{2} + 1,

(d) x^{6} − 4.

Problem 3: Let K_{1} and K_{2} be two subfields of a field K. The composite field K_{1}K_{2} of K_{1} and K_{2} is the intersection of all subfields of K containing both K_{1} and K_{2}. Let K_{1} and K_{2} be two extension fields of a field k contained in a field K.

(a) Prove that [K_{1}K_{2} : k] ≤ [K_{1} : k][K_{2} : k]. (In particular, K_{1}K_{2}/k is finite if K_{1}/k and K_{2}/k are both finite.)

(b) Prove that K_{1}K_{2}/k is algebraic if K_{1}/k and K_{2}/k are both algebraic.

Problem 4: Let K_{1} and K_{2} be two extension fields of a field k contained in a field K, and assume both are the splitting fields over k of certain polynomials in k[x].

Prove that K_{1} ∩ K_{2} is a splitting field over k. (Hint: Prove first that K_{1}K_{2} is a splitting field.)