Gauss was awarded a fellowship to the Collegium Carolinum by the Duke of Brunswick and attended from 1792 to 1795. Preceding this he went to the University of Gottingen from 1795 to 1798 and it was here that he had many mathematical discoveries that turned out to be rediscoveries in many cases due to the fact that he did not have access to a strong mathematical library. One of his more prominent advancements came when he revealed how to show that any regular polygon with a Fermit prime number of sides. It can be created using only a compass and a straight edge.    
    It can be argued that Gauss’s most productive and best year as a mathematician came in 1796 when he was able to: Invent modular arithmetic, create the quadratic reciprocity law, produce the prime number theorem, and conclude that positive integers can be represented as a sum of three triangular numbers. This year concluded with the publishing of even more great results.
    In 1798 Gauss completed his greatest work at the age of 21which was fundamental in consolidating number theory as a discipline in Disquistiones Arithmeticae, however the work was not published till 1801. After Gauss received criticism of his 1799 publication of the fundamental theorem of algebra which many people attempted to do before him, Gauss only published three more proofs. This is around the time when Gauss took his life focus away from pure mathematics as he figure that it only had limited claim in the real world. After working on planetary motion Gauss sought a position in astronomy and got one in 1807 as the Professor of Astronomy and Director of the Astronomical Observatory in Gottingen. This position he held the rest of his life. 
    When Gauss was 23 he predicted the position of Ceres with in an error of ½ a degree and from this he published a theory of celestial movement. This is  still important in astronomical computations and from this theory the Gaussian gravitational constant was developed and the method of least squares was revolutionized to minimize measurement error.
    In 1810 Gauss was asked and did with much enthusiasm a geodetic survey of the state of Hanover. This surveying eventually gave way to the development of Gausian or normal distribution from differential geometry dealing with curves and surfaces.
     In later years his work developed knowledge in the field of magnetism, which led to the invention of Kirchhoff’s circuit laws for electricity. With Wilhelm Weber Gauss built the first electromagnetic telegraph in 1833.  Before he died he would build a magnetic observatory and create a "magnetic club" to measure the earth’s magnetic field in many areas of the world.