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”

# Author Archives: Tair Galili

## 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”

## Roots of unity

After we’ve finally proved The Fundametal theorem of Galois Theory, we can officially say that we understand the correspondence, and we know exactly when it behaves as we want it to behave. So, it’s time to see some results! The first topic I want to discuss is related to the roots of unity. We’ve metContinue reading “Roots of unity”

## 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”

## Generators and Relations

In this post we are going to face the very last universal problem, it won’t be so hard though it won’t be as easy as the previous problem. However, the solution for the problem is great! we will use it a lot, since it can be really comfortable in some cases. Ok, let’s do it:Continue reading “Generators and Relations”

## The Free Group

So, in this post we will face the fifth universal problem. It’s really not difficult, though it will be extremely useful, and not just for our purposes! We will define the free group (over a given set). Since this one is really important, After solving the universal problem, I want to show some properties ofContinue reading “The Free Group”

## 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”

## Free product

We’ve already solved two universal problems! In this post, we will solve the third, which is not more complicated than the last problem. However, the next problem is pretty complicated and it will use what we’ll do here, so it’s important to understand this problem as good as possible. Ok, let’s not spend any moreContinue reading “Free product”

## Abelianization

After solving our first universal problem, It’s time for the second one. And as you may guess from the title, we are going to meet a new term which is called “Abelianization“. We’ll see exactly what does that mean: The universal problem Now the rules of the game are: We are given only one groupContinue reading “Abelianization”