Question

Let \(R\) be a UFD with field of fractions \(F, f, g \in R[x]\) nonzero of degrees \(n, m\), respectively. and \(\alpha_{1}, \ldots, \alpha_{n}\) and \(\beta_{1}, \ldots, \beta_{m}\) the roots of \(f\) and \(g\). respectively, in an extension field of \(F_{\text {, counted }}\) with multiplicities. (i) Prove: $$ \operatorname{res}(f, g)=\operatorname{lc}(f)^{m} \prod_{1 \leq i \leq n} g\left(\alpha_{i}\right)=(-1)^{n m} \operatorname{lc}(g)^{n} \prod_{1 \leq j \leq j m} f\left(\beta_{j}\right)=\operatorname{lc}(f)^{m} \operatorname{lc}(g)^{n} \prod_{\substack{1 \leq i

Short answer

Resultant \( \operatorname{res}(f, g) = \operatorname{lc}(f)^m \prod_{i=1}^n g(\alpha_i) = (-1)^{nm} \operatorname{lc}(g)^n \prod_{j=1}^m f(\beta_j) = \operatorname{lc}(f)^m \operatorname{lc}(g)^n \prod_{i,j} (\alpha_i - \beta_j) \).

Step-by-step solution

Understand the Resultant

The resultant of two polynomials \( f \) and \( g \), \( \operatorname{res}(f, g) \), is a key concept. It evaluates to zero if and only if \( f \) and \( g \) have a common root. Mathematically, it is defined via several equivalent expressions linking roots and leading coefficients.

Express Resultant with Roots

Assume \( f(x) = \operatorname{lc}(f)x^n + \cdots \) and \( g(x) = \operatorname{lc}(g)x^m + \cdots \). Then in an extension field, \( f \) and \( g \) have roots \( \alpha_1, \ldots, \alpha_n \) and \( \beta_1, \ldots, \beta_m \), respectively. One way to express the resultant is by evaluating \( g(x) \) at roots \( \alpha_i \) of \( f(x) \): \[ \operatorname{res}(f, g) = \operatorname{lc}(f)^m \prod_{i=1}^n g(\alpha_i) \] This product of evaluations combines polynomials via roots.

Switch Polynomials and Evaluate with Respective Roots

In another equivalent form, switch roles of \( f \) and \( g \), evaluating \( f(x) \) at roots \( \beta_j \) of \( g(x) \): \[ \operatorname{res}(f, g) = (-1)^{nm}\operatorname{lc}(g)^n \prod_{j=1}^m f(\beta_j) \] This evaluates products similarly, factoring in the symmetric role of \( f \) and \( g \) in resultants.

Use Symmetric Formula from Polynomial Theory

Combine both forms using properties of polynomial roots and resultant theories:\[ \operatorname{res}(f, g) = \operatorname{lc}(f)^m \operatorname{lc}(g)^n \prod_{i=1}^n \prod_{j=1}^m (\alpha_i - \beta_j) \]This uses symmetric product expressions of differences between roots of respective polynomials to compute the resultant's magnitude.

Verify Consistency of Formula

Both derived equations from switching roles and root differences should equate due to consistent theory in polynomial algebra. Comparison with given resultant forms confirms identity and demonstrates their equivalence, fulfilling part (i) of the problem statement.

Key concepts

Polynomial Roots

Understanding the roots of polynomials is pivotal in mathematics. A polynomial's roots are the solutions or values for which the polynomial equals zero. For instance, if you have a polynomial \( f(x) = 0 \), the roots \( \alpha_1, \alpha_2, \ldots \) are the values of \( x \) that satisfy this equation. In the context of the exercise, these roots play a crucial role, as the resultant of two polynomials relies on the evaluations of one polynomial at the roots of the other.

To visualize, consider a simple quadratic polynomial \( f(x) = ax^2 + bx + c \). Its roots can be found using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For more complex polynomials, roots might not be so straightforward, involving multiple terms or even requiring numerical methods for approximation.

• Roots are the values for which the polynomial evaluates to zero.
• They can be real or complex, and often appear in conjugate pairs.
• Roots are fundamental in determining the behavior of polynomials.

By understanding polynomial roots, we see how they influence the structure and resultant of polynomial equations.

Field of Fractions

The concept of a field of fractions comes up in many areas of algebra. Essentially, a field of fractions is a way to "expand" a given integral domain (like integers) into a field (like rational numbers). If you start with a Unique Factorization Domain (UFD), you can create a field of fractions to enable division by non-zero elements.

For example, think of integers, which form a ring. By extending them, you create the rational numbers, where any integer can divide another (except by zero). This extension allows us to better explore and unravel polynomial expressions.

• A field of fractions is built from a UFD to permit division.
• It expands our number system, allowing for more flexibility.
• This is vital for analyzing polynomials in terms of their roots and resultants.

A field of fractions equips us for deeper algebraic structures, enabling examination and manipulation of polynomials at a more complex level.

Unique Factorization Domain (UFD)

A Unique Factorization Domain is an upgraded concept from familiar integers. In a UFD, every element can be broken down uniquely into irreducible factors, similar to the prime factorization of integers. This property is crucial for understanding polynomial behavior and supports exploring concepts like resultants.

Consider the integers: each number can be uniquely factored into a product of prime numbers (e.g., 28 = 2\(\times\)2\(\times\)7). Similarly, in a UFD, polynomials can be factorized uniquely, a property that is foundational in ensuring that solutions like resultants are consistent and meaningful.

• UFDs allow unique decomposition into irreducible factors, ensuring consistency.
• They are essential in polynomial studies.
• The unique factorization assists in proving and expressing results in polynomial algebra.

The nature of UFDs makes them an ideal setting for studying polynomial equations and their roots, as seen in caluclations involving resultants.