The number theoretic functions equipped with Dirichlet convolution and prefix sum, form one of the most important topics in elementary number theory.In this text, we are going to study the number theoretic functions, however this is done from algebraic perspective by studying the Dirichlet ring, which they form under addition and Dirichlet convolution.Actually, one who gets the chance to study both elementary number theory and algebra in parallel will find how convenience and simple it is by...(read more)

Here comes the best known and most important application of Galois theory: to judge whether an equation is solvable by radicals.One of the most common misconception by folks is that Galois proved equation of degree greater than is not solvable. Since the proof had been given by Abel and Ruffini for near twenty years ago before Galois's work. In fact, Galois proved the sufficient and essential condition, that an equation related to a polynomial is...(read more)

In this text, we will introduce the Galois theory, which is a tool to correspond fields with groups under certain circumstances. This will empower us to study fields by studying their corresponding groups.A field automorphism of is a field isomorphism whose domain and image are both . We know there's , where the complex number conjugation...(read more)

What field is roots of in? They are obviously not in by Eisenstein's criterion or rational root theorem. And since we've already known are the solution of it, and it's in , so the roots are in . What a simple question.But what if I ask, is there any smaller field than that contains ? Consider ...(read more)

We have already been familiar with the fundamental theorem of arithmetic every since our elementary school, that for every positive integer can be decomposed into product of postitive prime power form , and the decomposition is unique.When it comes to the case of decomposing a polynomial in over , for a reduction of polynomial , it...(read more)