Problem 17 - Abstract Algebra: Theory and Applications

Question

Let \(F\) be a field such that \(\operatorname{char}(F) \neq 2 .\) Prove that the splitting field of \(f(x)=\) \(a x^{2}+b x+c\) is \(F(\sqrt{\alpha}),\) where \(\alpha=b^{2}-4 a c\)

Step-by-Step Solution

Answer

Answer: The splitting field of \(f(x) = ax^2 + bx + c\) is \(F(\sqrt{\alpha})\), where \(\alpha = b^2 - 4ac\).

Step 1: Find the roots of \(f(x)\) using the quadratic formula

Using the quadratic formula, we can find the roots of \(f(x)=ax^2 + bx + c\) as follows: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Step 2: Analyze the field extensions required for the roots

For the roots to be in the splitting field, we need to be able to adjoin any necessary field extensions to \(F\). In this case, the only non-trivial extension that may be necessary is \(\sqrt{\alpha}\), where \(\alpha = b^2 - 4ac\). If both roots are already in \(F\), then we can simply take \(F\) as the splitting field. Otherwise, we will need to adjoin \(\sqrt{\alpha}\) to \(F\), so our splitting field will be \(F(\sqrt{\alpha})\).

Step 3: Show that \(F(\sqrt{\alpha})\) contains both roots

First, observe that, by construction, \(F(\sqrt{\alpha})\) contains \(\sqrt{\alpha}\). To see that \(F(\sqrt{\alpha})\) contains both roots, consider the two cases: Case 1: At least one of the roots is not in \(F\). If one of the roots is not in \(F\), then the element \(\sqrt{b^2 - 4ac}\) must not be in \(F\). But then \(\sqrt{\alpha} = \sqrt{b^2 - 4ac}\) is not in \(F\), so we need to adjoin it to get our splitting field. Thus, both roots will be in \(F(\sqrt{\alpha})\). Case 2: Both roots are in \(F\). In this case, we don't need to adjoin anything to \(F\), as the roots are already in the field. Therefore, the splitting field is just \(F\), which is equal to \(F(\sqrt{\alpha})\) in this case.

Step 4: Show that \(F(\sqrt{\alpha})\) is indeed the smallest field containing both roots

It is clear that both roots of \(f(x)\) are in \(F(\sqrt{\alpha})\). Now we need to show that there is no proper subfield of \(F(\sqrt{\alpha})\) containing both roots. Since \(\alpha = b^2 - 4ac\) is directly related to the discriminant of the quadratic polynomial, any field containing both roots must also contain \(\sqrt{\alpha}\). But \(F(\sqrt{\alpha})\) is the smallest field containing \(F\) and \(\sqrt{\alpha}\) by definition, so there can be no proper subfield containing both roots. In conclusion, the splitting field of \(f(x) = ax^2 + bx + c\) is \(F(\sqrt{\alpha})\), where \(\alpha = b^2 - 4ac\).

Key Concepts

Quadratic Formula

The Quadratic Formula is a fundamental concept in mathematics used to find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\). This formula is particularly useful because it offers a direct solution when factoring is difficult or impossible.
The formula is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula requires the calculation of the discriminant \(b^2 - 4ac\), which determines the nature of the roots.
A quadratic equation can have:

• Two distinct real roots if the discriminant is positive.
• One real root if the discriminant is zero (also known as a repeated root).
• Two complex roots if the discriminant is negative.

The Quadratic Formula thus not only finds the roots but also gives insight into the characteristics of the solutions.

Field Extensions

Field Extensions are an advanced topic in algebra dealing with the expansion of a given field \(F\) to include additional elements, thereby creating a larger field. When solving polynomials, especially in cases where the roots are not initially in the field, extensions allow us to adjoin necessary elements.
For example, given a quadratic polynomial \(f(x) = ax^2 + bx + c\), if the discriminant \(b^2 - 4ac\) is not a perfect square within \(F\), we need to extend the field by adding \(\sqrt{b^2 - 4ac}\).
This process creates a new field, \(F(\sqrt{\alpha})\), where \(\alpha = b^2 - 4ac\).
Field extensions are essential in finding splitting fields, which are the smallest fields containing all roots of a polynomial.
Through field extensions, we can systematically explore and construct necessary fields that contain solutions to given polynomial equations.

Discriminant

The Discriminant is a specific term in the context of quadratic equations that helps determine the nature of the roots of the equation. It is represented by \(\Delta = b^2 - 4ac\) and is derived from the quadratic formula.
Understanding the discriminant value can give insights as follows:

• If \(\Delta > 0\), the equation has two distinct real roots. This indicates that the quadratic touches the x-axis at two separate points.
• If \(\Delta = 0\), there is exactly one real root, or the two roots are identical (a double root). This scenario means the vertex of the parabola lies on the x-axis.
• If \(\Delta < 0\), the quadratic has two complex roots, indicating the parabola does not intersect the x-axis.

The discriminant is crucial for discerning the number and type of solutions to quadratic equations and plays an essential role in field theory when discussing splitting fields and extensions.

Field Theory

Field Theory is a branch of algebra that focuses on the properties and structures of fields. A field is a set equipped with two operations, addition and multiplication, which satisfy certain properties such as distributivity, associativity, and commutativity.
Field Theory explores concepts like splitting fields and field extensions, which are important for understanding polynomials and their roots.
In the context of splitting fields, Field Theory helps us determine the smallest field where a polynomial can be broken down into its linear factors.
This includes understanding which elements need to be added to the original field to achieve the necessary expanded field, such as adding \(\sqrt{b^2 - 4ac}\) in cases where the quadratic’s roots are not initially available.
Field Theory not only aids in solving polynomial equations but also helps establish a framework for further exploration in algebra and number theory.