Properties of exponantiation of ordinals

In the last post, we worked pretty hard to define ordinals exponentiation, however, This hard work revealed some great properties: If is a successor then: If is a limit ordinal then; Those properties are great – the first two are pretty intuitive, and the third is also a pretty comfortable definition of . We canContinue reading “Properties of exponantiation of ordinals”

The Cyclotomic polynomial

After we’ve met the terms of roots of unity, primitive roots, and deduced a great identity, we are now finally ready to define the Cyclotomic polynomial of order . It is the polynomial where it’s roots are the primitive roots of unity of order : Let’s find out what’s so special about it: First, noteContinue reading “The Cyclotomic polynomial”

The Fundamental Theorem of Galois Theory

After meeting the term of a Galois extension and Artin’s lemma, we are ready to prove the fundamental theorem of galois theory! Recall that when is a galois extension, we can say a lot about the extension: is a normal and separable. is a splitting field of a separable polynomial. is a fixed field, thatContinue reading “The Fundamental Theorem of Galois Theory”

Galois correspondence

After meeting the term of an automorphism, and, of more importantly, the Galois group of an extension, we would like to learn a bit more about the connection between the group and the extension by definig a correspondece betwee sub-group and sub-fields. Let’s see how we can do such a thing: Sub-groups to sub-fields SupposeContinue reading “Galois correspondence”

Free product with amalgamation

We are now facing the fourth universal problem – the most challenging one yet! Don’t worry though, I’ll try to explain it as simple and organized as possible. Moreover, despite it being challenging, it is also the most rewarding problem, as we’ll see in the future. Let’s do it: The problem Now there are threeContinue reading “Free product with amalgamation”