Category Archives: Galois Theory

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”

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”

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”

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”

In the last post, we’ve met the strong term of a galois extension – We’ve seen 5 different definitions of it! Before I move on to the fundamental theorem of galois theory, I want to present Artin’s lemma – a strong lemma that will turn out to be pretty handy in the proof of theContinue reading “Artin’s lemma”

In the last post, we’ve met the Galois correspondence between sub-fields and sub-groups: However, we had two major problems: We don’t know if . So far, all we know is that . We don’t know if we can that and ? In other words, we don’t know when those maps are inverse maps of eachContinue reading “Galois extensions”

In the one of my previous posts, we’ve discussed about splitting fields of a polynomial . Those are fields where splits in, and they are the minimal fields with this property. We’ve also seen that the splitting field is unique up to isomorphism. We worked kind of hard trying to figure out In how manyContinue reading “Automorphisms”

In the last post I’ve introduced the term of a separable polynomial. In this post I want to dive deeper into the defintion and persent some properties and criteria of separable polynomials. Characteristic of a field My major plan for this post is to present and prove a criteria that will determine wether or notContinue reading “Separable Polynomial”

In the last post, We’ve seen ways to extend an existing field. In this post, I would like to do two things: Discuss about the degrees of extensions Bring the term of a splitting field and prove it’s uniqueness (up to isomorphism). Let’s begin Multiplicative Property Suppose that are three fields. I’ve already proved thatContinue reading “Splitting fields & degrees of extensions”

In the last post, We’ve seen how to extend a field using an irreducible polynomial. Here, I want to show another way to do so, however, it’s going to be from a different prespective. However, I would like to give a few reminders first from ring theory Prime Ideals and PID’s Suppose that is aContinue reading “Fields from Elements & Elements from Fields”