Invariant theory is the study of invariants of homogeneous polynomials. Perhaps the best known example of an invariant is the discriminant b^2 - 4ac of a quadratic polynomial. In this thesis, we...Show moreInvariant theory is the study of invariants of homogeneous polynomials. Perhaps the best known example of an invariant is the discriminant b^2 - 4ac of a quadratic polynomial. In this thesis, we will consider invariants of binary quintics, which are homogeneous polynomials in two variables of degree five. Given values for the invariants, we (re)construct a binary quintic that attains this tuple of values. We will also discuss the implementation of these methods in SageMath, a free open-source mathematics software system.Show less
Adèles and idèles are nowadays frequently used in theoretical algebraic number theory, for example in class field theory. For explicit computations however, people still use the classical ideals....Show moreAdèles and idèles are nowadays frequently used in theoretical algebraic number theory, for example in class field theory. For explicit computations however, people still use the classical ideals. In this thesis we define representations of adèles and idèles, enabling us to perform explicit computations in adèle rings and idèle groups. We also discuss two applications of our representations: computing the profinite Fibonacci graph and computing Hilbert class fields of imaginary quadratic number fields using Shimura’s reciprocity law. We implemented these representations as well as the applications in the computer algebra package SageMath.Show less