Exercise 13, § 3

Let $K$ be a field, $P(T, X) = a_0(T)X^n + a_1(T)X^{n-1} + \ldots + a_n(T)$ a polynomial in $K[T, X]$; suppose that: 1" the polynomial $a_0$ is not divisible by $T$; 2" each of the polynomials $a_1, ..., a_n$ is divisible by $T$; 3" the polynomial $a_n$ is not divisible by $T^2$. Show that under these conditions $P$, as polynomial of $K(T)[X]$, is irreducible (argue by contradiction, using Ex. 10).

\* 11) With the notation as in Ex. 11, show that if we take $g(X) = X$ and $h(X) = X^n + X + 1 \ (n \geq 2)$, the polynomial $(g(X)h(x) - h(X)g(x))/(X-x)$ of $K(x)[X]$ is irreducible (use Ex. 13).

\* 15) Give an example of a field K containing two subfields $K_1, K_2$ such that K is algebraic over $K_1$ and over $K_2$, but not over $K_1 \cap K_2$. (Take $K = \mathbf{Q}(T)$, $K_1 = \mathbf{Q}(T^2)$, and $K_2 = \mathbf{Q}(T^2 - T)$. Verify that $[K : K_1] = 2$, $[K : K_2] = 2$ and $K_1 \cap K_2 = \mathbf{Q}$. To do this note that an element $f(T)$ of $K_1 \cap K_2$ satisfies the conditions $f(-T) = f(T)$ and $f(1 - T) = f(T)$.)