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Algorithms for fields and an application to a problem in computer vision

This thesis is composed of several different parts. We start with an investigation of an important problem in computer vision. An appropriate mathematical modeling of this problem motivates a problem in invariant theory, the examination of the natural action of the group PGL_{m+1} x S_n on the set of n-point configurations (P^m_K)^n (for some infinite field K and some m, n in N). 
This in turn leads to an investigation of algorithms for fields. We develop an algorithm for the intersection of fields (in special cases) and a method for testing whether a field is algebraically closed in another field. Moreover, we give an algorithm for testing simpleness of a field extension as well as finding - if applicable - a generating element. The latter leads to a new proof of a generalized version of the Theorem of Luroth. 
Another approach to solve problems in field theory is given by the theory of cross-sections of rational maps. We provide a survey on cross-sections and give a criterion for their existence. This yields an algorithm for testing field membership. 
Finally, we come back to the examination of the natural action of the group PGL_{m+1} x S_n on the set of n-point configurations. We determine generators of the corresponding invariant field and investigate their separating properties.