Problem 8 - Algebra

Question

Im folgenden sind jeweils Polynome \(P, Q \in R[X]\) über einem Ring \(R\) gegeben. Untersuchen Sie, ob Polynome \(S, T \in R[X]\) mit \(P=S Q+T\) mit \(\operatorname{deg}(T)<\) \(\operatorname{deg}(Q)\) existieren, und berechnen Sie diese gegebenenfalls (bzw. begründen Sie, warum diese nicht existieren). (a) \(P=\overline{4} X^{4}+\overline{2} X+1, Q=\overline{3} X^{2}-X \in \mathbb{Z}_{8}[X]\). (b) \(P=X^{3}+2, Q=3 X^{2}+1 \in \mathbb{Q}[X]\). (c) \(P=3 X^{3}+2 X^{2}, Q=3 X^{2}+1 \in \mathbb{Z}[X]\). (d) \(P=6 X^{4}-2 X^{3}+3 X^{2}, Q=2 X^{2}+1 \in \mathbb{Z}[X]\). (e) \(P=\overline{3} X^{3}+\overline{2} X+\overline{1}, Q=\overline{6} X^{2}+X \in \mathbb{Z}_{8}[X]\).

Step-by-Step Solution

Answer

Each part (a-e) has solutions for polynomials \(S\) and \(T\) such that \(T\) fulfills the degree condition.

Step 1: Identify Degrees of Polynomials

For each pair of polynomials \(P\) and \(Q\), determine the degrees of \(P\) and \(Q\). This will help us identify whether \(S, T \in R[X]\) can exist such that \(P = SQ + T\) with \(\operatorname{deg}(T) < \operatorname{deg}(Q)\).

Step 2: Polynomial Division for Part (a)

Polynomials: \(P = \overline{4} X^{4} + \overline{2} X + 1\) and \(Q = \overline{3} X^{2} - X\) in \(\mathbb{Z}_{8}[X]\). Divide \(P\) by \(Q\): Evaluate consecutive terms of \(P\) by multiplying and subtracting corresponding terms of \(Q\) until the degree of the remainder is less than 2.

Step 3: Result of Part (a)

Performing the division, we find: \( P = (\overline{5} X^{2} + \overline{6})Q + (\overline{5}X + 1) \) which means \(S = \overline{5} X^{2} + \overline{6}\) and \(T = \overline{5}X + 1\) with \(\operatorname{deg}(T) < 2\).

Step 4: Polynomial Division for Part (b)

Polynomials: \(P = X^{3} + 2\) and \(Q = 3X^{2} + 1\) in \(\mathbb{Q}[X]\). Divide \(P\) by \(Q\): Align the same degrees, subtract to find new terms until the remainder has a lesser degree than that of \(Q\).

Step 5: Result of Part (b)

Resulting division gives: \( P = (\frac{1}{3}X - \frac{2}{9})Q + \frac{20}{9} \) which implies \(S = \frac{1}{3}X - \frac{2}{9}\) and \(T = \frac{20}{9}\) with \(\operatorname{deg}(T) < 2\).

Step 6: Polynomial Division for Part (c)

Polynomials: \(P = 3X^3 + 2X^2\) and \(Q = 3X^2 + 1\) in \(\mathbb{Z}[X]\). Perform division: Use leading terms of \(P\) and \(Q\), subtract to simplify, and repeat until rest meets degree condition.

Step 7: Result of Part (c)

Performing the operation provides: \( P = (X)Q + 1\times(X^2 - 1) \) making \(S = X\) and \(T = -X^2 + 1\) with \(\operatorname{deg}(T)<2\).

Step 8: Polynomial Division for Part (d)

Polynomials: \(P = 6X^4 - 2X^3 + 3X^2\) and \(Q = 2X^2 + 1\) in \(\mathbb{Z}[X]\). Dividing \(P\) by \(Q\): Understand and calculate remainder with smaller degrees until it parallels conditions.

Step 9: Result of Part (d)

Finalized division process results in: \( P = 3X^2Q - 3X^3 + 0\times(X^2 - 1) \), where \(S = 3X^2\) and \(T = 0\) with \(\operatorname{deg}(T)<2\).

Step 10: Polynomial Division for Part (e)

Polynomials: \(P = \overline{3}X^{3} + \overline{2}X + \overline{1}\) and \(Q = \overline{6}X^{2} + X\) in \(\mathbb{Z}_{8}[X]\). Analyze each potential coefficient division aligned till the correct remainder is reached.

Step 11: Result of Part (e)

After division calculation, the solution becomes: \(P = (X + \overline{7})Q + \overline{4} \) where \(S = X + \overline{7}\) and \(T = \overline{4}\) having \(\operatorname{deg}(T)<2\).

Key Concepts

Polynomial Degree

When working with polynomials, one fundamental concept you need to understand is the degree of a polynomial. The degree is the highest power of the variable present in the polynomial. For instance, in the polynomial \( X^3 + 2 \), the highest power of the variable \( X \) is 3, so the polynomial degree is 3.

Understanding the degree of polynomials is crucial when you perform operations like addition, subtraction, multiplication, and especially division. It helps to indicate the complexity and behavior of the polynomial when graphed.

• A polynomial is expressed as \( a_nX^n + a_{n-1}X^{n-1} + \, \ldots \, + a_1X + a_0 \), where the degree is \( n \) and \( a_n eq 0 \).
• For example: The polynomial \( \overline{4} X^4 + \overline{2} X + 1 \) has a degree of 4.

This concept is vital, especially when examining if polynomials \( P \), \( Q \), \( S \), and \( T \) fulfill the equation \( P = SQ + T \) with the condition \( \operatorname{deg}(T) < \operatorname{deg}(Q) \). This guarantees that the remainder has a lower degree than the divisor, which signifies a complete division.

Remainder Theorem

The remainder theorem provides a straightforward way to determine the remainder of a polynomial \( P(X) \) when divided by a linear divisor \( (X - a) \). The theorem states that the remainder of this division is equal to \( P(a) \), the polynomial evaluated at \( a \).

This theorem is particularly useful in polynomial division, as it allows for efficient calculation. However, in the context of dividing by non-linear divisors, such as those seen in the exercise, the concept of the remainder is more complex. Here, you would manually perform polynomial division until the degree of the remainder \( T \) is less than the degree of the divisor \( Q \).

• If dividing \( P \) by \( X - c \), then if \( P(c) = 0 \), \( (X - c) \) is a factor of \( P \).
• In the polynomial division process, the remainder is what remains once you cannot perform further division that results in an integer polynomial.

For practical applications, like in the problem provided, understanding how to obtain and interpret the remainder helps us ensure the division is executed correctly, with \( \operatorname{deg}(T) < \operatorname{deg}(Q) \).

Modular Arithmetic

Modular arithmetic is often described as "clock arithmetic," where numbers "wrap around" to 0 after reaching a certain value—the modulus. This type of arithmetic is particularly helpful when working within a ring of polynomials with coefficients from a finite field, like \( \mathbb{Z}_8 \).

In the exercise, polynomials like \( \overline{4} X^4 + \overline{2} X + 1 \) and \( \overline{3} X^2 - X \) are considered in \( \mathbb{Z}_8[X] \), meaning each coefficient is computed modulo 8. This ensures that every arithmetic operation result, whether addition, subtraction, or multiplication, remains within the bounds of 0 to 7.

• When dividing polynomials in \( \mathbb{Z}_8[X] \), keep track of each step to ensure results conform to modular rules.
• This creates a cyclical pattern in the outcomes, simplifying calculations in many contexts.

Understanding modular arithmetic is vital for solving problems involving polynomials over finite fields, as it underpins the logic of operations in these restricted domains.

Ring of Polynomials

A ring of polynomials is a set of polynomials that form a mathematical structure adhering to ring properties. These properties include closure under addition and multiplication, an additive identity (usually the zero polynomial), and a multiplicative identity (usually the polynomial 1 or \( X^0 \)).

In the realm of polynomials over a ring \( R \), denoted as \( R[X] \), each coefficient of the polynomials belongs to the ring \( R \). This allows for consistent and reliable application of polynomial operations within the defined structure. In our examples, we deal with polynomial rings over integers \( \mathbb{Z}[X] \), rationals \( \mathbb{Q}[X] \), and finite fields \( \mathbb{Z}_8[X] \).

• A polynomial ring extends the concept of a ring to include polynomial elements.
• Both addition and multiplication of polynomials are closed operations in a polynomial ring.

Understanding the structure of polynomial rings helps appreciate their robust properties and supports executing operations like those in the given exercise, ensuring solutions respect the ring's axioms.