transcendence degree

The transcendence degree  of a set $S$ over a field $K$, denoted $T_{S}$, is the size of the maximal subset $S^{\prime}$ of $S$ such that all the elements of $S^{\prime}$ are algebraically independent  .

The transcendence degree of a field extension $L$ over $K$ is the transcendence degree of the minimal  subset of $L$ needed to generate $L$ over $K$.

Heuristically speaking, the transcendence degree of a finite set  $S$ is obtained by taking the number of elements in the set, subtracting the number of algebraic elements in that set, and then subtracting the number of algebraic relations  between distinct pairs of elements in $S$.

Example 1 (Computing the Transcendence Degree).

The set $S=\{\sqrt{7},\pi,\pi^{2},e\}$ has transcendence $T_{S}\leq 2$ over $\mathbb{Q}$ since there are four elements, $\sqrt{7}$ is algebraic, and the polynomial  $f(x,y)=x^{2}-y$ gives an algebraic dependence between $\pi$ and $\pi^{2}$ (i.e. $(\pi,\pi^{2})$ is a root of $f$), giving $T_{S}\leq 4-1-1=2$. If we assume the conjecture that $e$ and $\pi$ are algebraically independent, then no more dependencies can exist, and we can conclude that, in fact, $T_{S}=2$.

Title transcendence degree TranscendenceDegree 2013-03-22 13:58:11 2013-03-22 13:58:11 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 12F20 transcendence degree of a set transcendence degree of a field extension