transcendensgrad

Transcendensgrad is a concept in abstract algebra, specifically in the study of field extensions. It measures how "transcendental" a field extension is, meaning how many elements are needed to generate the extension that are not roots of any polynomial with coefficients in the base field.

Let K be a field and L be an extension field of K. An element $\alpha \in L$ is algebraic over K if there exists a non-zero polynomial P(x) with coefficients in K such that P($\alpha$) = 0. If no such polynomial exists, $\alpha$ is transcendental over K.

The transcendensgrad of a field extension L/K, denoted by trdeg(L/K) or tr.deg.(L/K), is the smallest integer n such that there exist elements $\alpha_1, \alpha_2, \ldots, \alpha_n \in L$ for which L is algebraic over the field K($\alpha_1, \alpha_2, \ldots, \alpha_n$). Equivalently, it is the size of a maximal algebraically independent subset of L over K. An algebraically independent set {$\alpha_1, \ldots, \alpha_n$} is a set where no element is algebraic over the field generated by the others.

If trdeg(L/K) = 0, then L is algebraic over K. If trdeg(L/K) = n > 0, then there exists a subfield M of L such that K $\subseteq$ M $\subseteq$ L, trdeg(M/K) = n, and L is algebraic over M. This means that M is a purely transcendental extension of K, isomorphic to the field of rational functions in n variables over K, and L is an algebraic extension of M.

The transcendensgrad is an important invariant for field extensions and plays a crucial role in the classification of fields and in algebraic geometry, where it corresponds to the dimension of an algebraic variety.