In algebra and geometry, the Galois theory has a special application, f.e. to check the existence of solutions to polynomials in the form of roots. It is another question to find explicit roots. Real approximations (these are rational numbers) of solutions can easily be calculated. In number theory, integer and rational solutions are often of greater interest. A main question is also the distinction between algebraic and transcendental numbers.In 9 chapters, after an introduction, permutations and groups, polynomials, the transformation of polynomials, solutions to polynomials, algebraic and transcendental numbers, the Galois theory, applications in cubic and elliptic equations, and drawings with compass and ruler are discussed.