Problem 2 - A First Course in Abstract Algebra

Question

The function \(v\) for \(\mathrm{Z}[x]\) given by \(v(f(x))=(\) degree of \(f(x))\) for \(f(x) \in \mathrm{Z}[x], f(x) \neq 0\)

Step-by-Step Solution

Answer

The function returns the degree of a non-zero polynomial with integer coefficients.

Step 1: Understanding the Function

The function described, \( v(f(x)) \), is a function that takes a polynomial \( f(x) \) from the ring \( \mathrm{Z}[x] \) (which means polynomials with integer coefficients) and returns an integer representing the degree of \( f(x) \).

Step 2: Analyzing the Domain

The domain of this function is \( f(x) \in \mathrm{Z}[x], f(x) eq 0 \). This means we are interested in non-zero polynomials with integer coefficients. Remember, the degree is defined for polynomials that are not the zero polynomial.

Step 3: Degree of a Polynomial

The degree of a polynomial \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \) is the highest power of \( x \) with a non-zero coefficient \( a_n \). For example, the degree of \( 3x^2 + 5x + 2 \) is \( 2 \).

Step 4: Result from function

For any non-zero polynomial \( f(x) \), \( v(f(x)) \) will return the integer that represents its degree. If \( f(x) = 3x^2 + 5x + 2 \), then \( v(f(x)) = 2 \).

Key Concepts

Polynomials with Integer Coefficients

A polynomial is an expression consisting of variables and coefficients, wherein the basic operations of addition, subtraction, and multiplication are used. When we speak of **polynomials with integer coefficients**, we mean that each term of the polynomial has a coefficient that is an integer. This means every number multiplying the variable is a whole number, which can be positive, negative, or zero.
Let's consider the polynomial \( 4x^3 - 5x + 7 \). Here, the coefficients are 4, -5, and 7—all integers. This whole expression forms a polynomial with integer coefficients due to these whole number multipliers.
Such polynomials are well-suited for many algebraic operations and are fundamental in understanding polynomial functions' behavior, especially when we discuss areas like roots or factoring.

Ring of Polynomials

The term **ring of polynomials** refers to a set of polynomials equipped with two operations: addition and multiplication. Specifically, we often refer to \( \mathbb{Z}[x] \), which is the set of all polynomials with integer coefficients. This set forms a **ring** because it satisfies the ring properties:

• Closure under addition and multiplication: If you add or multiply two polynomials with integer coefficients, the result is also a polynomial with integer coefficients.
• Associative and distributive laws are satisfied, ensuring algebraic operations behave predictably.
• An additive identity (zero polynomial) exists, making sure the sum of any polynomial with the zero polynomial is the polynomial itself.

Rings are critical in algebra because they provide a framework that extends beyond simple number operations, allowing us to tackle more complex structural mathematics.

Non-zero Polynomial

A **non-zero polynomial** is one where at least one coefficient is non-zero, effectively meaning the polynomial is not equivalent to zero. If all coefficients of a polynomial are zero, then we have the zero polynomial, which has undefined degree.
For example, \( f(x) = 3x^2 + x + 5 \) is a non-zero polynomial since it has terms with non-zero coefficients (3, 1, and 5).
Non-zero polynomials are vital when discussing properties like the degree of the polynomial. When analyzing a polynomial's degree, which is the highest exponent with a non-zero coefficient, it becomes fundamental to ensure the polynomial is non-zero. This ensures the meaningful evaluation of its degree and prevents undefined results, which can arise if we mistakenly attempt to assess the degree of the zero polynomial.