Basics of Ring Theory -- Basics of Polynomial Theory -- Groebner Basis -- Polynomial Resultants -- Gauss-Jacobi Combinatorial Algorithm -- Local versus Global Positioning Systems -- Partial Procrustes and the Orientation Problem -- Positioning by Ranging -- From Geocentric Cartesian to Ellipsoidal Coordinates -- Positioning by Resection Methods -- Positioning by Intersection Methods -- GPS Meteorology in Environmental Monitoring -- Algebraic Diagnosis of Outliers -- Transformation Problem: Procrustes Algorithm II -- Computer Algebra Systems (CAS). While preparing and teaching 'Introduction to Geodesy I and II' to - dergraduate students at Stuttgart University, we noticed a gap which motivated the writing of the present book: Almost every topic that we taughtrequiredsomeskillsinalgebra,andinparticular,computeral- bra! From positioning to transformation problems inherent in geodesy and geoinformatics, knowledge of algebra and application of computer algebra software were required. In preparing this book therefore, we haveattemptedtoputtogetherbasicconceptsofabstractalgebra which underpin the techniques for solving algebraic problems. Algebraic c- putational algorithms useful for solving problems which require exact solutions to nonlinear systems of equations are presented and tested on various problems. Though the present book focuses mainly on the two ?elds,theconceptsand techniquespresented hereinarenonetheless- plicable to other ?elds where algebraic computational problems might be encountered. In Engineering for example, network densi?cation and robotics apply resection and intersection techniques which require - gebraic solutions. Solution of nonlinear systems of equations is an indispensable task in almost all geosciences such as geodesy, geoinformatics, geophysics (just to mention but a few) as well as robotics. These equations which require exact solutions underpin the operations of ranging, resection, intersection and other techniques that are normally used. Examples of problems that require exact solutions include; - three-dimensional resection problem for determining positions and orientation of sensors, e. g. , camera, theodolites, robots, scanners etc. , VIII Preface - coordinate transformation to match shapes and sizes of points in di?erent systems, - mapping from topography to reference ellipsoid and, - analytical determination of refraction angles in GPS meteorology.