Finding the Minimal Degree Monic Polynomial with Given Roots

In mathematics, particularly in the study of polynomials, one often encounters the challenge of constructing a polynomial with specific roots. This article focuses on the process of finding the minimal degree monic polynomial with given roots, specifically those involving both real and complex numbers. We will explore the specific case where the roots are given as 3 - 2i√5 and √3 - 1.

Understanding the Problem

The problem at hand is to find the minimal degree monic polynomial with integer coefficients that has 3 - 2i√5 and √3 - 1 as roots.

Roots and Polynomial Properties

A monic polynomial is one whose leading coefficient is 1. Polynomials with integer coefficients have the property that if they have complex roots, those roots must be conjugates of each other. Similarly, if a polynomial has rational coefficients, any irrational root must have its conjugate also as a root.

The Roots and Their Properties

Let's start by analyzing the first root: 3 - 2i√5.

Determining the Minimal Polynomial for a Complex Root

To ensure that the polynomial has integer coefficients, we will use the minimal polynomial of the root 3 - 2i√5. This polynomial will also have the conjugate root 3 2i√5. We can determine this polynomial as follows:

[ (x - (3 - 2isqrt{5})) (x - (3 2isqrt{5})) 0 ]

Expanding this expression:

[ x^2 - (3 - 2isqrt{5} 3 2isqrt{5}) x (3 - 2isqrt{5})(3 2isqrt{5}) 0 ]

This simplifies to:

[ x^2 - 6x 29 0 ]

Determining the Polynomial for a Rational Root

Next, let's consider the root √3 - 1. Since the polynomial must have rational coefficients, it must also have the conjugate root -√3 - 1. The polynomial for these roots can be determined as:

[ (x - (sqrt{3} - 1)) (x - (-sqrt{3} - 1)) 0 ]

Expanding this expression:

[ (x - sqrt{3} 1)(x sqrt{3} 1) x^2 - (sqrt{3} - 1 sqrt{3} 1)x (sqrt{3} - 1)(sqrt{3} 1) x^2 - 2x - 2 ]

Combining the Polynomials

To find the combined polynomial, we need to multiply the two minimal polynomials identified above:

[ (x^2 - 6x 29)(x^2 - 2x - 2) 0 ]

Expanding this product:

[ x^4 - 2x^3 - 2x^2 - 6x^3 12x^2 12x 29x^2 - 58x - 58 x^4 - 8x^3 41x^2 - 46x - 58 ]

Conclusion

The minimal degree monic polynomial with integer coefficients that has the roots 3 - 2i√5 and √3 - 1 is:

[ x^4 - 8x^3 41x^2 - 46x - 58 ]

This polynomial meets the criteria of having the specified roots and is the minimal degree monic polynomial with integer coefficients that satisfies this condition.

Additional Insights

Finding polynomials with specific roots involves understanding the properties of polynomials, particularly those related to the conjugates of roots. This problem demonstrates the importance of considering both complex and rational roots in constructing minimal polynomials.