20. Polynomials Over Fields and Properties

In this lecture, we will continue our discussion on polynomials over rings. We will see polynomials over fields and discuss how to divide polynomials over fields. We will also discuss factorization of polynomials and define the notions of reducible and irreducible polynomials.

This chunk introduces the concepts of polynomials over fields and outlines the main topics of the lecture. It sets the stage for discussing polynomial division and factorization. A polynomial over a field is a mathematical expression formed from variables and coefficients, where the coefficients are drawn from a field - a set equipped with two operations that generalizes the arithmetic of rational numbers.

Think of polynomials as recipes: the variables are the ingredients (like flour or sugar), and the coefficients are the specific amounts used in the recipe (like 2 cups of flour). Over fields, it’s like cooking where you have all the necessary ingredients in precise amounts, allowing you to create consistent and reliable dishes.

Let us start with the usual division of polynomials...until the degree of the remainder is less than the degree of the divisor.

This chunk explains the fundamental method for dividing polynomials, similar to how we divide numbers. The process involves multiplying the divisor by various terms to match the leading term of the dividend and then subtracting to find the remainder. This process continues until the degree of the remainder is less than that of the divisor.

Imagine you are cutting a cake (the polynomial) into equal slices (the divisor). Each time you cut, you remove a piece (subtract) and continue until you’re left with a piece too small to cut again (remainder).

Now, what can we say about the addition of polynomials and multiplication of polynomials over fields?...

This chunk discusses how the properties of polynomials behave differently over rings and fields. In fields, the product of polynomials retains a degree equal to the sum of their individual degrees, unlike in rings where this may not hold true due to potential cancellation of terms.

Think of fields as having a perfect slate: when you combine the ingredients (polynomials), you always know exactly how much you have. In contrast, with rings, you might sometimes 'lose' some ingredients through unexpected interactions, which can affect the final amount.

We are now trying to extend the division property...where the degree of your remainder polynomial will be strictly less than the degree of your divisor polynomial.

This chunk introduces the polynomial division theorem, stating that for any two polynomials, you can express one as the product of the other (divisor) and a quotient, plus a remainder that is smaller in degree than the divisor. This is an essential property when working with polynomials over fields, ensuring uniqueness.

Think of this as finding how many times you can stack boxes (divisor) into a larger box (dividend) until you can’t fit another box in without going over the limit of height (degree). The leftover space is your remainder.

The next thing we want to define is the GCD of polynomials over field...that d(x) is actually kind of a maximal possible common divisor of both a(x) and b(x).

This chunk defines the greatest common divisor (GCD) for polynomials similarly to integers - it is the highest degree polynomial that divides both without remainder. It emphasizes that the notion of 'greatest' is tricky with polynomials, hence the term 'maximal common divisor' is used.

Imagine you have two pieces of rope (polynomials). The GCD represents the longest common length you can cut from both ropes without leftover (remainder).

It turns out that the beautiful Euclid’s GCD algorithm...this algorithm based on repeated division.

This chunk explains how to find the GCD of two polynomials using an extension of Euclid’s algorithm, which involves repeatedly dividing the larger polynomial by the smaller one and updating the terms until the remainder is zero. This is methodical and ensures you can find the GCD precisely.

Think of measuring the longest common length between two pieces of wood: you keep cutting until you can no longer fit any cuts that are the same length; the length of your last cut before you cannot cut anymore is the GCD.

The next thing that once we have seen polynomial division...is not reducible in the sense I cannot factorize it out into other than the trivial factorization.

This chunk introduces factorization concepts, especially irreducible polynomials, which cannot be factored into polynomials of lower degrees. It highlights the importance of distinguishing trivial factors (like constants) from meaningful factorization.

Picture trying to break down a LEGO set: an irreducible structure is one that can’t be taken apart further without losing its essential shape. Any part that stands alone on its own without further breakdown is considered irreducible.

The next thing that we want to define is the factor theorem for polynomials over fields...such as the combination of your original polynomial say a(x) and b(x).

Finally, this chunk presents the factor theorem, which states that (x - α) is a factor of a polynomial if and only if substituting α into the polynomial yields zero. This is foundational in identifying polynomial roots and factors.

If you think of a polynomial as a path, then finding a root where it crosses the x-axis (where f(α) = 0) indicates a stopping point or a change in direction – that’s where the path can be divided.

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