Question

Let \(F\) be a field and \(n \in \mathbb{N}_{2}\). (i) Determine the discriminant disc \((f)=\) res \(\left(f, f^{\prime}\right)\) of a binomial \(f=x^{n}+b\), with \(b \in F^{\times}\), (ii) (Swan 1962) Determine the discriminant of a trinomial \(f=x^{n}+a x^{k}+b\), with \(1 \leq k

Short answer

(i) Disc of \(f(x) = x^n + b\) is \((-1)^{n(n-1)/2} n^n b^{n-1}\). (ii) Disc of \(x^n + ax^k + b\) generally requires examining specific \(a, b\), degrees, and field choices.

Step-by-step solution

Understanding the Problem

We're tasked with finding the discriminant (or resultant) of two polynomial setups involving a binomial and a trinomial over a field \(F\). The discriminant of a polynomial is related to the derivative and shows the properties such as roots being complex or multiple.

Compute the Derivative

The polynomial in part (i) is \(f(x) = x^n + b\). First, calculate the derivative: \(f'(x) = nx^{n-1}\).

Resultant of Binomial

The discriminant for the binomial \(f(x) = x^n + b\) is the resultant of \(f\) and its derivative \(f'\), denoted as disc \((f) = \text{res}(f, f')\). This involves finding a formula for the resultant of two polynomials, but importantly noting:\[\text{disc}(x^n + b) = (-1)^{n(n-1)/2} n^n b^{n-1}.\]

Set Up Trinomial and Derivative

Consider the trinomial \(f(x) = x^n + ax^k + b\). Calculate its derivative: \(f'(x) = nx^{n-1} + kax^{k-1}\).

Resultant of Trinomial

For the trinomial with \(f(x) = x^n + ax^k + b\), the discriminant can also be expressed using a formula connecting it to the nature of its roots and coefficients. However, the explicit formula can be highly complex, depending on \(k\) and \(n\). We denote its discriminant generally as an expression in terms of \(a, b, k,\) and \(n\).

Answering with Standard Forms

For a binomial \(f(x) = x^n + b\), the discriminant is straightforward and known. For the trinomial \(f(x) = x^n + ax^k + b\), while a general formula could be deduced for specific values (commonly low degrees are considered for simplification), Swan's 1962 work typically implies using called upon simplifying laws for specific fields or values.

Key concepts

Resultant of Polynomials

To better understand the discriminant of a polynomial, we need to delve into a concept called the resultant of polynomials. The resultant of two polynomials, say \(f(x)\) and \(g(x)\), is a scalar quantity that indicates whether the polynomials have a common root. If the resultant is zero, it implies that the roots of the polynomials overlap at least once.

When we talk about a polynomial \(f(x)\) and its derivative \(f'(x)\), the resultant tells us about the multiplicity of the roots of \(f(x)\). In simpler terms, it tells us if a polynomial has repeated roots. The discriminant of a polynomial \(f(x)\) is essentially the resultant of \(f(x)\) and \(f'(x)\).

• A non-zero discriminant implies that all roots are distinct
• If the discriminant is zero, there is at least one repeated root

Understanding resultants is crucial as it simplifies complex polynomial relationships, particularly in the context of field theory and polynomial derivatives.

Polynomial Derivatives

Polynomial derivatives are foundational in calculus and algebra since they reveal vital properties of the polynomial function that they are derived from. To compute the derivative of a polynomial \(f(x) = x^n + ax^k + b\), where \(b\) is a constant, we apply the power rule.

For example, the derivative of a term \(cx^m\) is \(mcx^{m-1}\). Thus:

• For \(x^n\), the derivative is \(nx^{n-1}\),
• For \(ax^k\), it is \(kax^{k-1}\),
• And for constant \(b\), the derivative is zero.

Based on this, the polynomial \(f'(x) = nx^{n-1} + kax^{k-1}\). Understanding these derivatives is essential as they are directly linked to the process of finding the disorder in the roots of polynomials, which is a key element in determining the discriminant.

Field Theory

Field theory plays a pivotal role in understanding polynomials. It provides a mathematical structure within which polynomials are analyzed. A field is a set of numbers with operations of addition, subtraction, multiplication, and division, excluding division by zero. Such a field must obey certain rules or axioms.

In this context, fields allow us to work with polynomials in a more generalized framework, whether real, complex, or finite fields. This becomes especially useful when we consider polynomials like \(f(x) = x^n + ax^k + b\). The coefficients \(a, b \in F^{\times}\) denote non-zero elements of the field \(F\), ensuring that we have the full spectrum of field operations.

• Fields enable the application of concepts such as derivation, resultant, and discriminant across various types of numbers.
• They help in simplifying polynomial arithmetic and factorization based on the properties of the field.

By utilizing field theory, mathematicians can define and work with polynomials over diverse numeric systems, facilitating deeper insights into algebraic structures and their behaviors.