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”

Splitting fields & degrees of extensions

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”

Fields from Elements & Elements from Fields

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”