19.2.9. Polynomials Over Rings

Now, the next thing that we want to discuss here is the polynomials over rings, which is a very important concept used in computer science. So, imagine you are given a ring with some abstract plus an abstract dot operation and imagine that your dot operation is commutative. Now, if I want to define a polynomial of degree n over this ring ℝ it will be of this form: p(x) = a_n ullet x^n + a_{n-1} ullet x^{n-1} + \ldots + a_0, and everything is similar to the notion of polynomials that we are familiar with.

This chunk introduces the concept of polynomials defined over abstract rings. In mathematics, a polynomial is an expression consisting of variables (like x) raised to whole number powers and multiplied by coefficients. Here, the coefficients come from a ring ℝ, which means they follow specific addition and multiplication operations defined in that ring. The polynomial's degree is defined by the highest power of x in the expression.

Think of a polynomial like a recipe that combines different ingredients (the coefficients) to create a dish depending on how many times (the degree) you want to use each ingredient. If you have 2 tablespoons of sugar and 3 cups of flour (like the coefficients) and you're making a cake (the polynomial), the highest amount of sugar you can use determines how sweet (the degree) your cake is.

Since my degree is n, my restriction will be that the coefficient a_n will be non-zero; all other coefficients are allowed to be 0, and each of these coefficients is from your set ℝ, and your plus operation and your dot operation are the ring operations; they are not integer plus or integer dot operations. So, now, if I consider the set of all polynomials of various degrees that are possible over this ring ℝ, I denote that infinite set by this notation ℝ[x].

This part explains that in a polynomial, for it to truly be of degree n, the coefficient of x raised to the power n must be non-zero. However, the other coefficients (for x to the powers less than n) can be zero. All coefficients belong to the ring ℝ. The notation ℝ[x] is used to denote the collection of all possible polynomials formed using coefficients from the ring. This set includes polynomials of every degree, making it infinite.

Imagine collecting cards of different types. Each card is marked with its rarity (the degree) but only some types of cards will exist in your collection. If each type needs at least one card to be counted as part of your collection (the non-zero coefficient), you could have many that don't make it to the main list (zero coefficients). So the set of your cards is like ℝ[x], including all your unique cards that could be used when playing games.

Now, let us see how we define the operations of polynomials over ring. So, we know how to add 2 integer polynomials. So, if I am given 2 polynomials, say a polynomial 2x^2 + 3x + 1 and say another polynomial 5x + 2, we can perform addition and multiplication by treating the coefficients normally, but now with respect to the abstract ring's operations.

This chunk discusses how to perform operations (addition and multiplication) on polynomials defined over rings, similar to how we add or multiply integer polynomials. The key difference is that the addition and multiplication are defined by the ring's operations, not necessarily normal arithmetic. For example, you would add coefficients using the ring's addition and multiply using the ring's multiplication. This is important to maintain the integrity of polynomials in different mathematical structures.

Think of it like baking two different types of cookies where the recipe calls for peculiar measurements that are modified (like you can add sugar in tablespoons or teaspoons). When adding two recipes (polynomials), you merge them based on how much of each ingredient fits in your baking pan (the operations defined by the ring) instead of just relying solely on conventional measurements.

For example, imagine I consider my ring to be ℤ and my plus operation is addition modulo 3 and my multiplication operation is multiplication modulo 3. Now, I am taking 2 arbitrary polynomials. One polynomial is of degree 2 (2x^2 + 2x + 1) and another polynomial of degree 1 (x + 2) belonging to this set ℤ[x]. Let us see the results of summation and multiplication.

In this section, a specific ring is defined where the operations are modulo 3. Two arbitrary polynomials are selected, and the operations of addition and multiplication demonstrate how coefficients combine following the rules of the defined ring. This showcases how polynomials behave differently under these operations compared to standard arithmetic operations.

Consider a clock that only goes up to 3 hours. If it’s 2 o’clock now and you’re told to add 2 hours, you would go to 1 o’clock instead of 4 o’clock because you wrap around (modulo operation). This mirrors how polynomials calculated under a ring filter results differently than traditional math since you’re bending to fit a rule.

So, basically what I am saying here is that if I consider the set of all possible polynomials over the ring, then even with the dot operation, if I consider this operation, then it will satisfy the; and of course, the summation of 2 polynomials the way I have defined here. Therefore, the overall algebraic structure will satisfy your ring axioms provided the dot operation in your ring ℝ is commutative.

This chunk concludes that polynomials over a ring, when certain operations (addition and multiplication) are defined as per ring axioms, still satisfy the properties that define a ring, provided that the multiplication in the ring is commutative. This means that when you multiply two polynomials, the order in which you multiply does not affect the result.

Imagine a team project where each member can work on any part of the project. No matter how you assign tasks (multiply polynomials), if everyone's efforts (the ring operations) are coordinated well, the end product will reflect all your combined work, independent of who did what first (commutative property).

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