1.13. Examples of Field Construction

Now let us see how exactly we can construct finite fields for any given pr where p is a prime number and this is very interesting because it says the following you give me any prime number p, I will show the existence of a finite field whose characteristic will be that prime number p. And the number of elements in the field will be pr and how exactly we construct such a field. So, for constructing such a field we will take the help of some irreducible monic polynomial where the coefficients are over ℤ and the degree of the polynomial will be r. Why r? Because r is also given as part of your input. So you are given a prime number p and value r, my goal is to show the existence of a finite field with characteristic p and with pr number of elements.

In this chunk, we discuss how finite fields can be constructed based on given prime numbers and their powers. A finite field is a mathematical structure that contains a finite number of elements, and they possess properties defined by the field axioms, such as closure, associativity, and the existence of additive and multiplicative identities. To construct such a field, we need a polynomial that is both irreducible (meaning it cannot be factored into the product of two non-trivial polynomials) and monic (leading coefficient is one). This polynomial will define the field's structure. The use of irreducible polynomials is crucial because they ensure that the field behaves correctly under the rules of polynomial arithmetic.

Imagine trying to create a unique recipe for a special dish using certain key ingredients (the irreducible polynomial). These ingredients must be mixed in specific ways (under polynomial operations) to create a dish (the finite field) that has a definite taste (finite number of elements) and characteristics (field properties). Just as the right combination of ingredients creates a dish with unique flavors, the right polynomial creates a field with unique mathematical properties.

So, to do that I am basically taking a monic irreducible polynomial with coefficients over ℤp whose degree is r, if you are wondering whether indeed such polynomials always exist for any given r and p, the answer is yes. Such polynomial always exists for every r and p and there are some standard methods for doing that; getting such polynomials but for some well known values of p and r such polynomials are publicly available. Now, my goal is to construct a field F, so, my set F will be the set of all polynomials with coefficients over ℤ modulo k(x). In other words, basically the set F is the collection of all polynomials of degree 0, degree 1, degree 2, degree 3 and up to degree r - 1 where the coefficients of the polynomial are from ℤ.

Here, we expand on the construction of the finite field F by clarifying what kind of polynomials we are dealing with. Each polynomial in this set can have its coefficients taken from integers in the range of 0 to p-1 (since we're working over ℤp). These polynomials can have degrees ranging from 0 up to r-1, which means we consider polynomials of varying complexity but confined in degree due to our earlier irreducible polynomial's degree limitations. This organization helps ensure the resultant polynomials behave according to the properties of a field.

Think of creating a building out of blocks. Each block represents a polynomial, and the size of the blocks corresponds to the degree of the polynomial—smaller blocks have lower degrees, while larger ones represent higher degrees. The choice of blocks and their assembly (polynomial operations) determines the stability and functionality of the building (the field) that we are constructing.

So this means degree is at most r - 1 where the coefficients are allowed from the set ℤp. Why I am saying it is at most r - 1, because since each of the coefficients are from the set ℤ and my ℤp have the elements from 0 to p – 1 that means I can have a polynomial where all the coefficients are 0 that means I can also have a polynomial which is the 0 polynomial. So it turns out that how many elements I can have; how many such polynomials I can have in my collection F. Since I can have each of the coefficients taken from the set ℤ, namely p each of the coefficients can take p possible values and each of them are picked independently that means it is not the case that the coefficient a depends on the coefficient is a, it is not the case that the coefficient a depends on the coefficient is a and a they are picked independently. So, I can say that from the product rule of counting there are pr number of possible polynomials in my collection F. So I have defined my collection F.

In this chunk, the conversation turns towards the operations defined on the polynomials within the finite field collection F. The degree constraint ensures that we are limited to polynomials of a manageable size, which aids in maintaining the structure of the field. Since each coefficient can independently take on p values, the overall number of distinct polynomials we can form becomes p raised to the power of r, representing all possible combinations. This aspect is crucial for understanding the size and structure of the finite field we're constructing.

Imagine a color palette containing p colors, and you have r slots to fill with these colors to create a unique painting (polynomial). Each slot can independently hold any of the p colors, so as long as you have r slots, the possible combinations of colors you can choose will be p^r, just like having so many variations in your final artwork.

Now I have to give the definition of the abstract plus operation and abstract dot operation. So, my plus operation here is defined to be the addition of polynomials where the coefficients are added as per ℤ namely p addition modulo p and then I take the resultant polynomial modulo the irreducible polynomial. So that will ensure that my resultant polynomial will have coefficients over ℤ and its degree will be at most r - 1, because the degree of k(x) is r. To begin with my a(x) and b(x) polynomials both those polynomials will have degree r - 1 and if I add any 2 polynomials of degree r - 1, at most I will still obtain a polynomial of degree at most r - 1.

This chunk presents how addition (the plus operation) and multiplication (the dot operation) are defined for the polynomials within the finite field. Specifically, we perform additions based on modular arithmetic, ensuring that the sum wraps around if it exceeds our modulus p. This operation maintains the polynomial's structure, allowing it to stay within the defined degree limit. Similarly, multiplication involves polynomial multiplication rules modulo the defining irreducible polynomial, ensuring that the resultant polynomial does not exceed the degree allowed by the field.

Consider a simple clock that runs on a 12-hour format. If it's 10 o'clock now and you add 5 hours, you will end up at 3 o'clock (addition modulo 12). In the same way, the addition of polynomials works within a confined degree, and if it exceeds a certain limit, you 'wrap around' using the irreducible polynomial for operations, just like how we use the clock's format to keep track of time.

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