Problem 3 - A Book of Abstract Algebra

Question

Find the root field of \(a(x)=\left(x^{2}-3\right)\left(x^{3}-1\right)\) over \(\mathbf{Q} .\) ANSWER The complex roots of \(a(x)\) are \(\pm \sqrt{3}, 1, \frac{1}{2}(-1 \pm \sqrt{3} i)\), so the root field is \(\mathbb{Q}\left(\pm \sqrt{3}, 1, \frac{1}{2}(-1 \pm \sqrt{3} i)\right)\). The same field can be written more simply as \(\mathbb{Q}(\sqrt{3}, i)\). Find the root field of \(x^{4}-2\), first over \(Q\), then over \(\mathbb{R}\).

Step-by-Step Solution

Answer

The root field over \(\mathbb{Q}\) is \(\mathbb{Q}(\sqrt[4]{2}, i)\); over \(\mathbb{R}\) is \(\mathbb{R}(\sqrt[4]{2})\).

Step 1: Find Roots of the Polynomial

The polynomial given is \(x^4 - 2\). To find its roots, solve the equation \(x^4 = 2\). The roots are \(x = \/pm \sqrt[4]{2}\) and \(x = \/pm i\sqrt[4]{2}\). These roots are complex since \(i = \sqrt{-1}\).

Step 2: Determine the Root Field over \(\mathbb{Q}\)

Over the field \(\mathbb{Q}\), the root field will include all roots of the polynomial \(x^4 - 2\). These are \(\pm \sqrt[4]{2}\) and \(\pm i\sqrt[4]{2}\). Therefore, the root field over \(\mathbb{Q}\) is \(\mathbb{Q}(\sqrt[4]{2}, i)\).

Step 3: Determine the Root Field over \(\mathbb{R}\)

Over the real numbers \(\mathbb{R}\), the complex numbers \(\pm i\sqrt[4]{2}\) cannot be included, so the field will only include the real roots \(\pm \sqrt[4]{2}\). The root field over \(\mathbb{R}\) is then \(\mathbb{R}(\sqrt[4]{2})\).

Key Concepts

Complex Roots

Complex roots are solutions to polynomial equations that include the imaginary unit, denoted by the symbol \( i \). An imaginary number is a number that when squared gives a negative result. The common imaginary unit \( i \) satisfies the equation \( i^2 = -1 \). This means complex roots include terms with \( i \), outputting numbers that can describe phenomena which cannot be resolved on the real number line alone.
For example, in the polynomial \( x^4 - 2 \), some solutions are \( \pm i\sqrt[4]{2} \), demonstrating the presence of imaginary components. Understanding complex roots is crucial for solving higher degree polynomials where the solutions cannot all be real numbers.

Polynomial Roots

A polynomial is a mathematical expression consisting of variables and coefficients combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation. The roots of a polynomial are the values of the variable that make the polynomial equal to zero.
To identify the polynomial roots of expressions like \( x^4 - 2 \), you solve it for \( x \) by finding values where the polynomial expression returns zero. In this case, roots include both real numbers \( \pm \sqrt[4]{2} \) and complex numbers, demonstrating that a polynomial can have diverse types of solutions. Understanding the full set of roots, including those which involve complex numbers, provides a complete picture of the polynomial’s behavior.

Quadratic Field

A quadratic field is a field extension of the rational numbers \( \mathbb{Q} \), constructed by adding the square root of a number that is not a perfect square within \( \mathbb{Q} \). These fields have the form \( \mathbb{Q}(\sqrt{d}) \), where \( d \) is an integer and \( d \) is not a perfect square.
For example, the root field of the polynomial \( x^2 - 3 \) involves adding \( \sqrt{3} \) to \( \mathbb{Q} \). Such fields are useful for handling mathematical expressions and solving equations where roots involve radicals—expressions with square roots that cannot be simplified within the rational numbers. Understanding quadratic fields allows for a deeper exploration of number systems and their extensions beyond the rational numbers.

Field Extension

Field extension refers to expanding a field with additional elements that satisfy a polynomial equation without a solution within the original field. Simply put, it is the process by which a new field is created by adding roots of polynomials to an existing field.
When we add \( i \) (the square root of \(-1\)) and \( \sqrt{2} \) (or \( \sqrt[4]{2} \) for \( x^4 - 2 \)) to \( \mathbb{Q} \) to form \( \mathbb{Q}(\sqrt{2}, i) \) or \( \mathbb{Q}(\sqrt[4]{2}, i) \), we perform a field extension. This enables solving and explaining complex equations that rational numbers alone cannot tackle. Field extensions are essential in algebra for constructing new number systems where certain polynomial equations become solvable.