Question

Suppose \(K\) is a field of characteristic zero, \(F\) an irreducible monic polynomial in \(K[X]\) of degree \(n>0 .\) Let \(L\) be a splitting field of \(F\), so \(F=\prod_{i=1}^{n}\left(X-x_{i}\right), x_{i} \in L\) Show that the \(x_{i}\) are distinct. (Hint: Apply Problem \(1.51\) (c) to \(G=F_{X} ;\) if \((X-x)^{2}\) divides \(F\), then \(G(x)=0\).)

Short answer

Question: Prove that if \(F\) is an irreducible monic polynomial over a field \(K\) of characteristic zero, and \(L\) is a splitting field of \(F\), then the roots of \(F\) in its splitting field are distinct. Answer: The roots of an irreducible monic polynomial \(F\) in its splitting field are distinct, since our assumption of a repeated root led to a contradiction with the irreducibility of \(F\).

Step-by-step solution

Understand the given information

We are given \(K\) a field of characteristic zero and \(F\) an irreducible monic polynomial in \(K[X]\) of degree \(n > 0\). We are also given \(L\), a splitting field of \(F\), and \(F\) can be written as $$F = \prod_{i=1}^{n} (X - x_i), x_i \in L$$ Our goal is to show that the roots \(x_i\) are distinct.

Compute the derivative of F

Find the derivative \(F_X\): $$F_X = \frac{dF}{dX} = \frac{d}{dX}\left(\prod_{i=1}^n (X - x_i)\right)$$

Apply the product rule to the derivative

Using the product rule for derivatives when differentiating the product of terms, we have, $$F_X = \sum_{j=1}^n (X - x_j)' \prod_{i\neq j} (X - x_i) = \sum_{j=1}^n \prod_{i\neq j} (X - x_i)$$

Show that if there is a repeated root, it leads to a contradiction in the irreducibility of F

Assume there exists a root (let's call it \(x_k\)) that is repeated, meaning it appears at least twice in the factorization of \(F\). Then \((X - x_k)^2\) divides \(F\). Let's evaluate \(F_X\) at \(x_k\): $$F_X(x_k) = \sum_{j=1}^n \prod_{i\neq j} (x_k - x_i)$$ Since \((X - x_k)^2\) divides \(F\) and \(F_X\) contains the product of the linear factors of F, we have that \(F_X(x_k) = 0\). Now, we have a common root (\(x_k\)) in both \(F\) and \(F_X\). According to Problem 1.51(c), for the field \(K\) with characteristic zero, a polynomial is irreducible if and only if its polynomial and its derivative do not share any common roots. But we have found a common root \(x_k\), leading to a contradiction. So our assumption that there exists a repeated root must be wrong.

Conclude that the roots of F are distinct

Since our assumption that there is a repeated root led to a contradiction, we can conclude that the roots \(x_i\) of the polynomial \(F\) are distinct.

Key concepts

Polynomial Irreducibility

Understanding the irreducibility of a polynomial, particularly within a field of characteristic zero, is a fundamental concept in algebra. An irreducible polynomial is one that cannot be factored into polynomials of lower degree that have coefficients in the same field. This means that the polynomial does not have any non-trivial divisors.

In the context of the given exercise, the polynomial F is monic (its leading coefficient is 1) and irreducible in K[X], which implies that it has no factors within the field K other than trivial factors (which are units in the field and multiples of the polynomial itself).

Irreducibility is a key property when considering the existence of distinct roots. If a polynomial could be factored into smaller degree polynomials over the same field, then it would indicate the existence of roots within the field. However, being irreducible enforces that, if roots exist, they do so in an extension field, and it's related to the polynomial's inability to be factored over its coefficients' field. This is why the concept of a splitting field, like L, is introduced to accommodate these roots.

Field of Characteristic Zero

A field of characteristic zero is one where the sum of one added to itself any number of times never results in zero unless it's added zero times. In simpler terms, there is no natural number n such that adding the multiplicative identity to itself n times equals the additive identity of the field.

Fields of characteristic zero are critical when discussing polynomial functions and their derivatives. One of the properties of such fields is that a non-zero polynomial and its derivative will not have any common factor, assuming the polynomial is non-constant. This is because the coefficients of the polynomial are not subject to the modular arithmetic that would result from a field with non-zero characteristic. Essentially, the usual rules of calculus apply, including the fundamental concept that the derivative of a polynomial of degree n is a polynomial of degree n-1.

This property allows the argument used in the exercise, where a non-zero derivative indicates that there cannot be a repeated root. If a root were repeated, it would also be a root of the derivative, which is a contradiction for fields with characteristic zero. This is a subtlety that hinges entirely on the nature of the field's characteristic.

Distinct Roots

The term distinct roots refers to the roots of a polynomial where no root is repeated; each root represents a unique solution to the polynomial equation F(x) = 0. In a splitting field, a polynomial is factored completely into linear factors, which allows us to examine the roots directly.

In the splitting field L of the polynomial F, we are guaranteed that F can be represented as the product of linear factors (X - x_i). Showing that these roots are distinct is a fundamental aspect of understanding the structure of splitting fields and the behavior of polynomials over these fields.

The derivative test is one way to demonstrate the distinctness of roots in fields of characteristic zero. It is based on the fact that for a root to be repeated, it must be a root of both the original polynomial and its derivative. As proved in the steps provided for our exercise, this cannot happen for irreducible polynomials in such a field, because a common root with the derivative would indicate reducibility, which is not the case for F. Hence, within the splitting field L, all roots x_i must be distinct.