Why Finding Explicit Roots is Not Practical

Oct 13, 2025 by pynkocean.com 124 views

Introduction

In abstract algebra, the concept of a splitting field is crucial in understanding the properties of polynomials. Given a polynomial $f(x)$ with coefficients in a field $F$ , the splitting field of $f(x)$ over $F$ is the smallest field extension of $F$ in which $f(x)$ splits into linear factors. In this article, we will explore the splitting field of the polynomial $f(x) = x^3 - x^2 + 1$ over the rational numbers $\mathbb{Q}$ .

Background

To begin, let's recall some basic definitions. A field extension $L/F$ is a pair of fields $F \subseteq L$ such that $L$ is a vector space over $F$ . The degree of the extension, denoted by $[L:F]$ , is the dimension of $L$ as a vector space over $F$ . A polynomial $f(x)$ with coefficients in $F$ splits in a field extension $L/F$ if it can be factored into linear factors in $L$ . The splitting field of $f(x)$ over $F$ is the smallest field extension of $F$ in which $f(x)$ splits.

The Polynomial $f(x) = x^3 - x^2 + 1$

The polynomial $f(x) = x^3 - x^2 + 1$ is a cubic polynomial with coefficients in $\mathbb{Q}$ . We are interested in finding the splitting field of $f(x)$ over $\mathbb{Q}$ . To do this, we need to find the roots of $f(x)$ .

Roots of $f(x)$

The roots of $f(x)$ are the values of $x$ that satisfy the equation $f(x) = 0$ . Unfortunately, finding the explicit roots of $f(x)$ is not straightforward. In fact, the roots of $f(x)$ are not rational numbers, and they cannot be expressed in terms of radicals.

Why Finding Explicit Roots is Not Practical

Finding the explicit roots of $f(x)$ is not practical for several reasons. Firstly, the roots of $f(x)$ are not rational numbers, which means that they cannot be expressed as fractions of integers. Secondly, the roots of $f(x)$ cannot be expressed in terms of radicals, which means that they cannot be expressed using square roots, cube roots, or other roots.

Finding the Splitting Field

Given that finding the explicit roots of $f(x)$ is not practical, we need to find an alternative approach to finding the splitting field of $f(x)$ . One approach is to use the concept of a minimal polynomial.

Minimal Polynomial

A minimal polynomial is a polynomial that has a specific property. Specifically, a minimal polynomial is a polynomial that has a root in a field extension $L/F$ and has the property that it is irreducible over $F$ . In other words, a minimal polynomial is a polynomial that cannot be factored into smaller polynomials over $F$ .

The Minimal Polynomial of $f(x)$

The minimal polynomial of $f(x)$ is a polynomial that has a root in the splitting field of $f(x)$ over $\mathbb{Q}$ and has the property that it is irreducible over $\mathbb{Q}$ . To find the minimal polynomial of $f(x)$ , we need to find a polynomial that has a root in the splitting field of $f(x)$ over $\mathbb{Q}$ and is irreducible over $\mathbb{Q}$ .

Finding the Minimal Polynomial

To find the minimal polynomial of $f(x)$ , we can use the concept of a field extension. Specifically, we can use the fact that the splitting field of $f(x)$ over $\mathbb{Q}$ is a field extension of $\mathbb{Q}$ .

The Field Extension $\mathbb{Q}(\alpha)$

Let $\alpha$ be a root of $f(x)$ . Then the field extension $\mathbb{Q}(\alpha)$ is a field extension of $\mathbb{Q}$ that contains $\alpha$ . We can use the fact that $\mathbb{Q}(\alpha)$ is a field extension of $\mathbb{Q}$ to find the minimal polynomial of $f(x)$ .

The Minimal Polynomial of $f(x)$

The minimal polynomial of $f(x)$ is a polynomial that has a root in the field extension $\mathbb{Q}(\alpha)$ and is irreducible over $\mathbb{Q}$ . To find the minimal polynomial of $f(x)$ , we can use the fact that $\mathbb{Q}(\alpha)$ is a field extension of $\mathbb{Q}$ .

The Minimal Polynomial of $f(x)$ is $x^3 - x^2 + 1$

The minimal polynomial of $f(x)$ is $x^3 - x^2 + 1$ . This polynomial has a root in the field extension $\mathbb{Q}(\alpha)$ and is irreducible over $\mathbb{Q}$ .

The Splitting Field of $f(x)$

The splitting field of $f(x)$ is the smallest field extension of $\mathbb{Q}$ in which $f(x)$ splits. To find the splitting field of $f(x)$ , we need to find a field extension of $\mathbb{Q}$ that contains all the roots of $f(x)$ .

The Field Extension $\mathbb{Q}(\alpha, \beta, \gamma)$

Let $\alpha$ , $\beta$ , and $\gamma$ be the roots of $f(x)$ . Then the field extension $\mathbb{Q}(\alpha, \beta, \gamma)$ is a field extension of $\mathbb{Q}$ that contains all the roots of $f(x)$ . We can use the fact that $\mathbb{Q}(\alpha, \beta, \gamma)$ is a field extension of $\mathbb{Q}$ to find the splitting field of $f(x)$ .

The Splitting Field of $f(x)$ is $\mathbb{Q}(\alpha, \beta, \gamma)$

The splitting field of $f(x)$ is $\mathbb{Q}(\alpha, \beta, \gamma)$ . This field extension contains all the roots of $f(x)$ and is the smallest field extension of $\mathbb{Q}$ in which $f(x)$ splits.

Conclusion

Q: What is the splitting field of $x^3-x^2+1$ over $\mathbb{Q}$ ?

A: The splitting field of $x^3-x^2+1$ over $\mathbb{Q}$ is $\mathbb{Q}(\alpha, \beta, \gamma)$ , where $\alpha$ , $\beta$ , and $\gamma$ are the roots of $x^3-x^2+1$ .

Q: Why is the splitting field of $x^3-x^2+1$ important?

A: The splitting field of $x^3-x^2+1$ is important because it is the smallest field extension of $\mathbb{Q}$ in which $x^3-x^2+1$ splits. This means that the splitting field of $x^3-x^2+1$ contains all the roots of $x^3-x^2+1$ and is the smallest field extension of $\mathbb{Q}$ that contains all the roots of $x^3-x^2+1$ .

Q: How do you find the splitting field of $x^3-x^2+1$ ?

A: To find the splitting field of $x^3-x^2+1$ , you need to find the roots of $x^3-x^2+1$ . Once you have found the roots, you can use the fact that the splitting field of $x^3-x^2+1$ is the smallest field extension of $\mathbb{Q}$ that contains all the roots of $x^3-x^2+1$ .

Q: What is the minimal polynomial of $x^3-x^2+1$ ?

A: The minimal polynomial of $x^3-x^2+1$ is $x^3-x^2+1$ . This polynomial has a root in the field extension $\mathbb{Q}(\alpha)$ and is irreducible over $\mathbb{Q}$ .

Q: Why is the minimal polynomial of $x^3-x^2+1$ important?

A: The minimal polynomial of $x^3-x^2+1$ is important because it is the smallest polynomial that has a root in the field extension $\mathbb{Q}(\alpha)$ and is irreducible over $\mathbb{Q}$ . This means that the minimal polynomial of $x^3-x^2+1$ is the smallest polynomial that contains all the roots of $x^3-x^2+1$ .

Q: How do you find the minimal polynomial of $x^3-x^2+1$ ?

A: To find the minimal polynomial of $x^3-x^2+1$ , you need to find a polynomial that has a root in the field extension $\mathbb{Q}(\alpha)$ and is irreducible over $\mathbb{Q}$ . Once you have found such a polynomial, you can use the fact that it is the minimal polynomial of $x^3-x^2+1$ .

Q: What is the relationship between the splitting field and the minimal polynomial of $x^3-x^2+1$ ?

A: The splitting field of $x^3-x^2+1$ is the smallest field extension of $\mathbb{Q}$ that contains all the roots of $x^3-x^2+1$ , and the minimal polynomial of $x^3-x^2+1$ is the smallest polynomial that has a root in the field extension $\mathbb{Q}(\alpha)$ and is irreducible over $\mathbb{Q}$ . This means that the splitting field of $x^3-x^2+1$ contains all the roots of the minimal polynomial of $x^3-x^2+1$ .

Q: How do you use the splitting field and the minimal polynomial of $x^3-x^2+1$ in practice?

A: In practice, you can use the splitting field and the minimal polynomial of $x^3-x^2+1$ to find the roots of $x^3-x^2+1$ . Once you have found the roots, you can use the fact that the splitting field of $x^3-x^2+1$ contains all the roots of $x^3-x^2+1$ to find the splitting field of $x^3-x^2+1$ .

Q: What are some common applications of the splitting field and the minimal polynomial of $x^3-x^2+1$ ?

A: Some common applications of the splitting field and the minimal polynomial of $x^3-x^2+1$ include:

Finding the roots of $x^3-x^2+1$
Finding the splitting field of $x^3-x^2+1$
Finding the minimal polynomial of $x^3-x^2+1$
Using the splitting field and the minimal polynomial of $x^3-x^2+1$ to solve equations involving $x^3-x^2+1$

Q: What are some common mistakes to avoid when working with the splitting field and the minimal polynomial of $x^3-x^2+1$ ?

A: Some common mistakes to avoid when working with the splitting field and the minimal polynomial of $x^3-x^2+1$ include:

Assuming that the splitting field of $x^3-x^2+1$ is the same as the minimal polynomial of $x^3-x^2+1$
Assuming that the minimal polynomial of $x^3-x^2+1$ is the same as the splitting field of $x^3-x^2+1$
Not using the correct field extension when finding the splitting field of $x^3-x^2+1$
Not using the correct polynomial when finding the minimal polynomial of $x^3-x^2+1$