Question

The aim of this exercise is to show that Corollaries \(6.17\) and \(6.21\) remain essentially true over arbitrary (commutative) coefficient rings \(R\). Let \(f, g \in R[x]\) nonzero of degrees \(n, m\), respectively. such that \(r=\operatorname{res}(f, g) \in R\) is nonzero. (i) Show that therc exist polynomials \(s, t \in R(x)\) such that \(s f+t g=\operatorname{res}(f, g)\). Hint: Apply Corollary \(6.21\) to the generic case where the coefficients of \(f\) and \(g\) are indeterminates, and then map them to the actual coefficients in \(R\). (ii) Now assume that \(f\) and \(g\) are monic. Prove that there exist polynomials \(s, t \in R[x]\) with deg \(s<\) \(\operatorname{deg} g\) and deg \(t<\operatorname{deg} f\) satisfying \(s f+t g=1\) if and only if res \((f, g)\) is a unit in \(R\).

Short answer

Existence of polynomials-s, t can be shown using the corollary, while the unit condition arises from invertibility.

Step-by-step solution

Understanding the Problem Statement

We are given nonzero polynomials \(f, g \in R[x]\) with degrees \(n\) and \(m\) respectively and \(r = \operatorname{res}(f, g)\) is nonzero. We aim to show the existence of polynomials \(s, t \in R(x)\) such that \(s f + t g = \operatorname{res}(f, g)\). Additionally, when \(f\) and \(g\) are monic, we need to demonstrate the existence of such \(s\) and \(t\) with specific degree conditions if and only if \(r\) is a unit.

Applying Corollary 6.21

The corollary using indeterminate coefficients suggests the existence of polynomials \(S, T\) in \(Q[x]\) such that \(S f + T g = \operatorname{res}(f, g)\) for a generic coefficient set. Through a homomorphism, map this result onto polynomials over \(R\) with specific coefficients, confirming that such \(s, t\) exist for \(R[x]\).

Mapping to Specific Coefficients

Considering \(f\) and \(g\) over a ring \(R\) with their real coefficients, utilize a polynomial ring homomorphism to map the indeterminate coefficient polynomials from the corollary to the given polynomials \(f\) and \(g\). This establishes the existence of such \(s\) and \(t\) in \(R(x)\) as \(rs(f,g)\) holds for indeterminate cases.

Showing the Unit Condition for Monic Polynomials

When \(f\) and \(g\) are monic, show that if \(\operatorname{res}(f, g)\) is a unit, \(|R|=1\) implies there are \(s, t \in R[x]\) with the degree condition \(\deg s < \deg g\) and \(\deg t < \deg f\) such that \(s f + t g = 1\). Conversely, if such \(s\) and \(t\) exist, then \(\operatorname{res}(f, g)\) must be a unit as \(s f + t g = \pm 1\), showcasing invertibility.

Key concepts

Commutative Rings

A commutative ring is a fundamental structure in algebra that extends the concept of a ring where the multiplication operation is commutative. In simple terms, this means that for any two elements \(a\) and \(b\) in the ring \(R\), the equation \(a \cdot b = b \cdot a\) holds true.

Such rings can be as straightforward as the set of integers \(\mathbb{Z}\), or more complex like polynomial rings \(R[x]\), which are formed by considering polynomials with coefficients in a ring \(R\). A crucial aspect of commutative rings in polynomial resultants is that they allow us to apply various algebraic properties, such as distributing multiplication over addition, and the commutative nature of multiplication ensures that order doesn’t matter, allowing for flexible manipulation of terms.

• Closure and Identity: These rings have an additive identity (0) and a multiplicative identity (1).
• Associativity and Distributivity: They obey laws of associativity for addition and multiplication, and multiplication distributes over addition.

Commutative rings form the basis upon which many algebraic tools are developed, including the examination of polynomial behaviors and properties, such as those found in this exercise.

Monic Polynomials

Monic polynomials are a special class of polynomials where the leading coefficient, the coefficient of the highest degree term, is 1. This property simplifies many algebraic processes and ensures the polynomial is in a more standardized form.

For example, the polynomial \( f(x) = x^3 + 3x^2 + 2 \) is monic because the coefficient of the \(x^3\) term is 1. Monic polynomials play a key role in the exercise, especially when dealing with resultants and ensuring certain conditions are met.

• Simplification: Working with monic polynomials often simplifies calculations because they remove the need to factor out non-unity coefficients.
• Standard Form: They provide a standardized way of presenting polynomials, especially useful when comparing polynomials.

Understanding monic polynomials helps in checking the invertibility conditions of resultants. Specifically, when polynomials are monic, it becomes easier to establish the necessary relationships between polynomial degrees and units in the ring, as demonstrated in the exercise.

Polynomial Homomorphism

Polynomial homomorphism involves mapping a polynomial from one ring to another, preserving the ring operations of addition and multiplication. This is particularly useful when transforming theoretical results into practical instances. For our exercise, it plays a critical role.

Using homomorphism, polynomials with indeterminate coefficients are mapped to polynomials with specific coefficients in a ring \(R\). This allows us to extend generic results to actual scenario instances in the ring \(R[x]\).

• Maintaining Operations: A homomorphism ensures that if you perform operations on polynomials in one structure, the results remain consistent when mapped to another structure.
• Translating Results: By translating general algebraic results into specific cases, we can derive meaningful conclusions about polynomial relationships in different rings.

In application, this method allows the flexible study of polynomial properties, ensuring that key results derived under certain assumptions apply universally across various coefficient sets. This provided the pathway for verifying the existence of polynomials \(s\) and \(t\) in \(R(x)\).

Algebraic Degree Conditions

Algebraic degree conditions refer to constraints placed on the degrees of polynomials involved in certain algebraic expressions. In this exercise, these conditions become crucial when establishing relationships between polynomials \(s\), \(t\), \(f\), and \(g\).

Specifically, the requirement for polynomials \(s\) and \(t\) to have degrees less than \(g\) and \(f\) respectively, ensures that when combined with \(f\) and \(g\), non-trivial solutions are obtained for the polynomial resultant equation. These conditions are derived from the degrees of the polynomials involved.

• Polynomial Degree: The degree of a polynomial is the highest power of the variable within it, serving as a primary measure of its "size" and complexity.
• Resultant Derivation: The first part of the exercise deals with finding polynomials \(s\) and \(t\) such that \(s f + t g = 1\), under specific degree constraints.

Adhering to algebraic degree conditions is crucial for establishing involvements and mappings between rings and their polynomial expressions. This ensures a meaningful relationship that results in a valid polynomial expression holding in the given ring space.