Exercise 6, § 17

Let D be a field with center K of degree $m^2$ over K. Let $x$ be an element of D of degree $n >1$ over K, and let $f(X)$ be its minimal polynomial over K.

a) Prove that a monic polynomial in D[X] (VIII, p. 11) of degree $< n$ cannot be right divisible by all polynomials $X-txt^{-1}$ for $t\in D^*$ (observe that the monic generator of the ideal $\cap (D[X] (X-txt^{-1}))$ has coefficients in K).

b) Let $u(X), v(X), w(X)$ be three monic polynomials in D[X] such that $u(X) =$ $v(X)w(X)$ and $X-x$ is a right divisor of $u(X)$ but not of $w(X)$. Write $w(X) =$ $q(X) (X-x) +t$ with $t\in D^*$. Prove that $X-txt^{-1}$ is a right divisor of $v(X)$.

c) Deduce from a) and b) that there exist $n$ elements $t_1, . . . , t_n$ of D such that we have

(observe that $f(X)$ is right divisible by all polynomial $X-txt^{-1}$).

d) Deduce that Trd$_{D/K}(x)$ (resp. Nrd$_{D/K}(x)$) is the sum (resp. product) of $m$ elements of the form $txt^{-1}$ (apply Proposition 8 of VIII, p. 347).