A Galois Field is actually a corn field owned by any person named Galois! Too obvious? Alright, that was a joke. Anyway, the word ‘field’ is being used in the mathematical context here. Évariste Galois was a mathematical prodigy who laid strong foundations for abstract algebra. His collected works contain important ideas that have had far-reaching consequences for nearly all branches of mathematics, thus etching his name in mathematics forever. Unfortunately, he died at a tender age of 20. The work he did as a teenager is now being used by mathematicians around the world for their doctorate studies and related research work. Prodigy indeed! Just so we are clear, this blog post has nothing to do with corn fields. Well then, what else can it be about?
What is a field?
In abstract algebra, we tend to talk about grouping things together, and that group is expected to behave in a certain way. The reason for doing this is because we want to understand the behavior of a group of things without analyzing every single thing. In a way, a field is a concept that groups together few elements and associates them with some properties.
More formally, a field is a commutative ring that contains a multiplicative inverse for every nonzero element. A ring, in turn, is an algebraic structure that generalizes the arithmetic operations of addition and multiplication. It means that for every non-zero element in the group, there exists an element which can be multiplied with it to get the identity element as the result (which is 1 in the case of multiplication). Equivalently, a field is a ring whose non-zero elements form an Abelian group under multiplication. A group is said to be Abelian if it satisfies the law of commutativity, which means that regardless of the order you use to operate on two elements, the result should be the same. For example, a+b should be the same as b+a.
What is a Galois field?
The person in this picture is Galois. Now that we put a face to this whole discussion, let’s go further. Fields can be divided into two types; infinite fields and finite fields. The infinite fields contain infinite elements. For example, the set of real numbers is a field with infinite elements. We don’t have much control over these fields as such. Fields with finite elements are the ones that contain finite number of elements, obviously. These finite fields are called Galois fields. They are used extensively in number theory, coding theory, cryptography, error correction, quantum mechanics, etc. As far the real life applications are concerned, finite fields are much more interesting because of the way they behave depending on how we formulate them.
How is it useful?
You will be surprised to know how often you interact with Galois fields, even if you are not a mathematician. The world, as it is today, is built on 0s and 1s. Computers, transistors, internet, machinery, cryptography, and many more fields are built entirely on 0s and 1s. As we all know, the number system with 0 and 1 is called binary system. But did you know that this binary system is actually a Galois field? Some of the most fundamental algorithms in modern times have been built using the properties of Galois fields. If you are interested, you should definitely read up more on Galois fields. It’s pretty fascinating!
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Perpetual Enigma 