Imagine that you have a very persistent piano teacher insisting that you study notes and practice scales for three years before you are allowed to listen to or play any real music. How is that going to affect your level of inspiration? Are you going to attend every lesson with passion or practice absolutely ignited with energy? Abstract algebra is like piano playing. You can kill your inspiration and motivation spending years on formalism before seeing the beauty of the subject. This book is written with the intent that every chapter should contain some real music, matters which involve practice of the notes and scales in a surprising and unexpected way. It is an attempt to include a lot of non-trivial and fun topics in an introductory abstract algebra course. Having inspiring goals makes the learning easier. The topics covered in this book are numbers, groups, rings, polynomials and Gröbner bases.

Knowledge of linear algebra and complex numbers is assumed in some examples. However, most of the text is accessible with only basic mathematical topics such as sets, maps, elementary logic and proofs.

Gröbner bases are usually not treated at an undergraduate level. My feeling four years ago when including this topic in the syllabus at Aarhus was one of hesitation. I was afraid that the material would be too advanced for the students. It turned out that the students liked the concrete nature of the material and enjoyed the non-trivial computations with polynomials.