Becker, Eberhard (PhD)Hunde, Natea2018-07-172023-11-042018-07-172023-11-042017-03http://etd.aau.edu.et/handle/123456789/8966This thesis is devoted to a study of real holomorphy rings in R(x) and elliptic function _elds R(x)(p f(x)); f is a monic polynomial of degree 3 with no multiple roots in C over R and sums of powers in R(x). The objects of study are real valuation rings, real holomorphy rings and real places in the rational function _elds R(x) and then investigate their extension to elliptic function _elds which are quadratic extension of R(x). We write down the full list of real valuation rings to give the description of holomorphy ring in both _elds and _nally we study applications to sums of powers.enThe Real Holomorphy Ring and SumsThesis