Diophantine problems related to cyclic cubic and quartic fields

We are interested in solving the congruences f3+g3+1≡0(modfg) and f4−4g2+4≡0(modfg) in polynomials f,g with rational coefficients. Moreover, we present results of computations of all integer points on certain one parametric curves of genus 1 and 3, related to cubic and quartic fields, respectively. Our approach is based on Gröbner basis techniques and we do numerical experiences based on it. We mainly deal with the case degf≤2 and prove that there are no new families of cyclic cubic nor cyclic quartic fields. In case of cyclic quartic fields we obtained a new polynomial that was not discovered by Balady and Washington [2], it is given by t4+7890798742t3−37333446t2+38618t+1.