Exercise 7, § 9

Let K be a field of characteristic 0, a, β, 5 three elements of an algebraic closure of K such that $[K(\alpha):K] = 2,\ [K(\beta):K] = 3,\ [K(\xi):K] = n > 1$; denote by a' (resp. $\beta', \beta''$) the unique conjugate $\neq a$ of a (resp. the conjugates $\neq \beta$ of β). a) Show that $[K(\alpha, \xi):K] = n$ if $a \in K(\xi)$, $[K(\alpha, 5):K] = 2n$ otherwise. b) Show that $[K(\beta, 5):K] = n$ if $\beta \in K(\xi)$, $[K(\beta, 5):K] = 2n$ if $\beta \in K(\xi)$ and just one of $\beta', \beta''$ belongs to $K(\xi)$ and finally $[K(\beta, \xi):K] = 3n$ if none of the three elements $\beta, \beta', \beta''$ belongs to $K(\xi)$. c) If d is the discriminant of the minimal polynomial of β, show that $K(\beta, \beta') = K(\beta, \sqrt{d})$. d) Show that $K(\beta, \beta') = K(\beta - \beta')$. e) If $n = 2$ and $a \notin K(\xi)$, show that $K(\alpha, \xi) = K(\alpha + \xi)$. f) Show that $K(\alpha, \beta) = K(a + \beta)$. g) Show that $K(\alpha, \beta) = K(\alpha \beta)$ except when $a = aj$ and $\beta^3 = b$, where a, b are elements of K and $j^3 = 1$.