22.1. Construction of Finite Field with 9 Elements

Let us do some warmup and see how exactly we construct finite fields. So we will see a construction of a finite field with 9 elements. I denote set F which is a collection of these 9 polynomials.

In this section, we begin by focusing on how to construct a finite field consisting of 9 elements. This field is represented as set F, which contains a collection of polynomials. Here, 'finite' implies that the field has a limited number of elements, specifically 9 in this case.

Think of a finite field like a small storage box that can only hold a certain number of items – in this case, 9 polynomials. Just as you can only fit a limited number of toys in a box, a finite field only contains a limited number of elements.

These are basically polynomials of degree 0 and degree 1 where the coefficients are from ℤ and remember ℤ is the set { 0, 1, 2 }.

The polynomials in this finite field are of two specific degrees: degree 0 (which is a constant) and degree 1 (which is linear). The coefficients of these polynomials are taken from the set of integers ℤ, specifically the elements {0, 1, 2}. This choice of coefficients restricts the polynomials to just 3 possible values, leading to the construction of our finite field.

Imagine you are making different shapes using only 3 colors: red, blue, and green. Each combination of colors that you use to paint shapes can be thought of as a polynomial, limited by the fact you can only use these three colors.

If I consider the operation of polynomial addition over the set ℤ [x], then it satisfies the closure property.

The closure property means that when you perform an operation (like addition) on elements in a set (like our polynomials), the result is still an element of that same set. In this case, if you pick any two polynomials from the set and add them, the result will also be a polynomial that belongs to the same set F.

Imagine you have a bag containing apples. If you take two apples from the bag and combine them, you still have apples in the bag. Similarly, adding two polynomials yields another polynomial in set F, maintaining the overall integrity of the set.

It turns out that with respect to the operation of polynomial multiplication, the closure property is not satisfied.

While addition works well, multiplication poses a challenge. When multiplying two polynomials, the result may produce a polynomial that is of a higher degree than allowed in set F. For example, multiplying two degree 1 polynomials can yield a degree 2 polynomial that doesn't fit into the defined set of degree 0 and 1 polynomials.

Imagine a scenario where you are trying to stack building blocks. If you combine too many blocks (polynomial degrees), you might end up with a tower that is too tall and exceeds the height limit you initially set (degree restriction).

I am going to define a modified addition and multiplication operation, where I will be doing all the addition and multiplication of the polynomial as I was doing earlier, but my resultant answer will be computed modulo this polynomial (x2 + 1).

To solve the problem of closure in multiplication, we introduce a modified operation that involves taking results modulo a specific polynomial, in this case, (x² + 1). This ensures that any result obtained remains within the limits of our defined set F by always reducing the outcome using this polynomial.

Think of this operation as setting rules in a game. If your score exceeds a certain limit, you reset to a base score. Similarly, with the modulo operation, we reset the results of polynomial operations to maintain them within our finite field.

Now, I would be interested to check whether this collection F indeed satisfies the axioms of field as well, with respect to the addition and multiplication operation modulo this irreducible polynomial.

The next step is to verify that our modified operations satisfy all the field axioms, such as associativity, commutativity, identity elements, and inverses for addition and multiplication. This means checking that every element in our field has an inverse (another element that will sum or multiply to yield the identity element, 0 or 1 respectively).

Imagine a board game where every player must have a pair of matching tokens to play (the inverses). If everyone has matching tokens, the game has structure and fairness, similar to how fields must operate under specific rules.

It turns out that indeed each non-zero element of this set F has a multiplicative inverse.

In our finite field F, every non-zero polynomial has a corresponding polynomial that can multiply with it to yield the multiplicative identity element, which is 1 in this case. For instance, we find that the inverse of polynomial 2 is also polynomial 2, as when multiplied yield 1 under the modulo operation.

Consider a buddy system where everyone pairs up. If everyone can find their pair (inverse), the system works effectively. Just like in our finite field, finding pairs of polynomials ensures the field's structure remains intact.

If you take the various powers of 2x + 1 and compute the powers as per the modified multiplication operation, then you get basically all the non-zero elements of this collection F.

In finite fields, a single element can generate all other non-zero elements through its powers. In this case, the element 2x + 1 serves as a generator, allowing us to obtain every non-zero polynomial when raised to successive powers. This property highlights the cyclic nature of the group formed by the field's non-zero elements.

Think of a recipe that uses a base ingredient to create various dishes. Just as that ingredient can be combined in different ways to yield a multitude of meals, the polynomial 2x + 1 can combine in various ways to generate all other polynomials in our field.

--