Problem 4 - A Book of Abstract Algebra

Question

For each of the following polynomials \(p(x)\), find a number \(a\) such that \(p(x)\) is the nimum polynomial of \(a\) over \(\mathbb{Q}\) : (a) \(x^{2}+2 x-7\) (b) \(x^{4}+2 x^{2}-1\) (c) \(x^{4}-10 x^{2}+1\)

Step-by-Step Solution

Answer

(a)\(-1 + \sqrt{8}\), (b)\(\pm\sqrt{-1+\sqrt{2}}\), (c)\(\pm \sqrt{5+2\sqrt{6}}\) or \(\pm \sqrt{5-2\sqrt{6}}\).

Step 1: Introduction of Minimal Polynomial

The minimal polynomial of a number \(a\) over \(\mathbb{Q}\) is the non-zero monic polynomial of least degree with rational coefficients that has \(a\) as a root. Therefore, \(p(x)\) must have \(a\) as a root and no polynomial of smaller degree satisfies this condition with rational coefficients.

Step 1: Solve Polynomial Equation for (a)

To find the number \(a\), solve the polynomial equation \(x^2 + 2x - 7 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=2\), and \(c=-7\). This gives us roots: \(x = \frac{-2 \pm \sqrt{4 + 28}}{2}\), which simplifies to \(x = -1 \pm \sqrt{8}\), so \(a = -1 + \sqrt{8}\) because the polynomial is monic.

Step 2: Solve Polynomial Equation for (b)

For the polynomial \(x^4 + 2x^2 - 1\), set \(y = x^2\), turning it into \(y^2 + 2y - 1 = 0\). Solve using the quadratic formula: \(y = \frac{-2 \pm \sqrt{2^2 + 4}}{2}\), which simplifies to \(y = -1 \pm \sqrt{2}\). Thus, \(x = \pm \sqrt{-1 + \sqrt{2}}\). Any such value for \(x\) is a candidate for \(a\).

Step 3: Solve Polynomial Equation for (c)

The polynomial \(x^4 - 10x^2 + 1\) can also be transformed by setting \(y = x^2\), resulting in \(y^2 - 10y + 1 = 0\). Solving this with the quadratic formula gives \(y = \frac{10 \pm \sqrt{100 - 4}}{2}\). Simplifying gives \(y = 5 \pm 2\sqrt{6}\). This translates to \(x = \pm \sqrt{5 + 2\sqrt{6}}\) or \(x = \pm \sqrt{5 - 2\sqrt{6}}\). Choose any such \(x\) for \(a\).

Step 5: Confirming Minimal Polynomial Property

In each case, the polynomials give values that satisfy their respective equations, and since these are monic equations of the lowest degree with rational coefficients having the roots, the polynomials are minimal.

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool used to find the roots of any quadratic equation. When dealing with a quadratic polynomial of the form \( ax^2 + bx + c = 0 \), the roots can be calculated using the formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula gives us the two possible values of \( x \) which satisfy the equation.

• "\( b^2 - 4ac \)" is known as the discriminant. It tells us about the nature of the roots.
• If the discriminant is positive, we have two distinct real roots.
• If it is zero, there's exactly one real root (a double root).
• If it is negative, the roots are complex and not real.

For example, in the given problem, the discriminant \(4 + 28\) is positive, leading to two real roots: \( x = -1 \pm \sqrt{8} \).

Polynomial Roots

Roots of a polynomial are values of \( x \) that make the polynomial equal to zero.
Finding the roots is essentially solving the equation \( p(x) = 0 \). For polynomials like the ones given in the exercise, the goal was to identify values \( x=a \) where the polynomial will evaluate to zero.

• The roots found are often critical in defining the minimal polynomial over a specific domain, such as the rational numbers.
• In the quadratic formula example, the irrational number \( \sqrt{8} \) necessitates the polynomial to be minimal due to rational coefficient restrictions.

Monic Polynomial

A monic polynomial is a polynomial whose leading coefficient, the coefficient of the term with the highest degree, is 1. This is significant in determining minimal polynomials because:

• Monic polynomials have a standardized form, simplifying comparison and calculations.
• They help define the minimal degree for a polynomial that can have a given number as a root, especially when rational coefficients are required.
• For example, the polynomial \( x^2 + 2x - 7 = 0 \) is monic. The leading coefficient is 1, making it suitable for expressing a minimal polynomial.

Rational Coefficients

Rational coefficients mean that all the coefficients in a polynomial are rational numbers, i.e., numbers that can be expressed as the quotient of two integers. This restriction is important when discussing minimal polynomials because:

• It ensures the polynomial can be expressed in terms that fit precisely within the rational field \( \mathbb{Q} \).
• Having rational coefficients eliminates the possibility of any irrational component influencing the polynomial's expression.
• This is crucial when formally defining a minimal polynomial over \( \mathbb{Q} \), as irrational coefficients would necessitate a completely different recursive relationship than arithmetic within the rationals.