Galois Theory - Introduction

Welcome to the first post of Galois theory! I am just going to say it right now, this series of posts, and this theory, requieres some pre-knowledge. What I mean by that is that the reader has to be familiar with linear algebra (or at least be familiar with vector spaces), which is a decent requirement since linear algebra is one of the first courses that is being taught almost at every scientific degree. The next requirement is not so trivial, you have to be familiar with groups. The main idea of Galois theory is to use a connection between groups and fields (which is called Galois correspondence) in order to learn about structres of fields and their sub-fields. I have three goals for this post:

Give a definition and an example of a Group
make some progress and introduce the definition of a ring
After we have those definitions, we will finally be able to define and understand what is a field.

Before all of that, I just want to give some motivation for this topic. Two of the most famous reults of Galois theory:

Proof that there is no formula for an equation of degree 5 and above.
Proof that states that 3 problem that people tried to solve over 2000 years are just unsolveable.

I will talk about the results in more detail in later posts, but for now, I just want you to think about these results, for example, one question that I ask is why 5? why this number, this specific number is the upper bound, why not 6? or 8? how can we proof such a thing? with my current tools I have no idea how to do so.

One note worth mentioning is that Galois Theory was developed by a teen! his name was Évariste Galois. He died at age 20 from wounds suffered in a duel. This man’s life were really short but still he was able to solve two major problems as a teen. If you ask me, that is one the craziest stories I have ever heared. Can you imagine what happened if he wouldn’t die? How many more things he would have accomplished, and how math would look today.

If I have to describe Galois theory in one word, that word would be magic. As we progress and dive deep to the theory we will see how magical it is, not to mention it’s special background, which makes it even more special. Ok, I think it is time to dive into the water, Let’s begin!

Groups

A Group $(G,\cdot)$ is a set $G$ equipped with an operation $(\cdot)$ . This operation takes two element of the set, and outputs one element from the set.

(\cdot):G\times G \to G

However, this structure must obey specific rules:

Closure: for every $a,b\in G$ , $a\cdot b$ is also in $G$ .
Associativity: $a\cdot (b\cdot c) = (a \cdot b)\cdot c$
Identity element: there is an elemet ( $1_G$ ) that satisfies: $1_G \cdot g = g \cdot 1_G = g$ for every $g\in G$ . That basically means that this element has ‘no influence’ on all the elements of the group under the operation
Inverse element: every element $g\in G$ has another element $g^{-1} \in G$ such that $g^{-1 }\cdot g = g \cdot g^{-1} =1_G$ .

And thats it! if a set satisfies all the above then it is called a Group. Let’s look at an example. Consider the set of integeres $\mathbb{Z}$ where the opertaion is the regular sum (+) as we know it. That means that for every two integers $a,b\in\mathbb{Z}$ the operation is: $a\cdot b = a+b$ , that is a little wired at first look, but you get used to it when you understand that $\cdot$ is just a symbol and not multiplication.

It is not very hard to see that the set with the operation satisfies the conditions and that $(\mathbb{Z},+)$ is indeed a Group.

There is one more definition I want to present here: if the operation satisfies $a\cdot b = b\cdot a$ then the group is said to be abelian. The example above, is an example of an abelian group.

Groups are going to play a big role from now on, so If you are not familiar with groups, and there are no posts here on groups yet, I suggest reading about groups. There are some great sources to learn from out there across the web!

Rings

After we defined what group are, in order to understand what a field is, we have to go through rings first. Rings are also an algebraic structure, which is a little more complicated then a group.

A Ring $(R,+,\cdot)$ is a set $R$ equipped with two operaitons. But those operations act in a special way:

$(R,+)$ is an abelian group
$(R,\cdot)$ is a almost a group (monoid) it fulfills all the group axioms but one : an element does not need to have an inverse.

However, that is not all, those operations also communicate with each other:

$x\cdot (y+z) = x\cdot y + x\cdot z$
$(y+z)\cdot x = y\cdot x + z\cdot x$

You may be familiar with this property, It is called the Distributive property. Quick example will be the ring of integers $(\mathbb{Z},+,\cdot)$ where $"+"$ is just the normal addition operation and $"\cdot"$ is the normal multiplication.

Notice one more thing, The monoid $(R,+)$ can be a group. If it is indeed a group, where the group is abelian, we get a strucutre caled field.

Fields

Finally, we can define what a field is. As we saw, a field is just a special case of a ring. In other words, we can define a field as trio $(F,+,\cdot)$ where:

$(F,+,\cdot)$ is a ring.
$(F,+),(F-\{0\},\cdot)$ are both abelian group.

The most famous field almost anyone is familiar with without even realizing is the field of real numbers, $(\mathbb{R},+,\cdot)$ . It can easilly be checked that it is indeed a field.

Field inner structure is not as interesting as ring inner structure, we will see it in later post when I’ll talk about Ideals, the interesting thing that we are going to do, is look at field and it’s subfields, and how a field and a subfield are actually forming a Vector space! The idea presented here is going to be our starting point, and from it, we can develop a beautiful theory. We will discuss all of the above in the upcoming posts, but for an introduction, I think that’s going to be enough. See you in the next one!

Galois Theory – Introduction

Groups

Rings

Fields

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Groups

Rings

Fields

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