examples of minimal polynomials

Note that $\sqrt[4]{2}$ is algebraic over the fields $\mathbb{Q}$ and $\mathbb{Q}(\sqrt{2})$. The minimal polynomials for $\sqrt[4]{2}$ over these fields are $x^4-2$ and $x^2-\sqrt{2}$, respectively. Note that $x^4-2$ is irreducible over $\mathbb{Q}$ by using Eisenstein’s criterion and Gauss’s lemma (http://planetmath.org/GausssLemmaII) (see this entry (http://planetmath.org/AlternativeProofThatSqrt2IsIrrational) for more details), and $x^2-\sqrt{2}$ is irreducible over $\mathbb{Q}(\sqrt{2})$ since it is a quadratic polynomial and neither of its roots ($\sqrt[4]{2}$ and $-\sqrt[4]{2}$) are in $\mathbb{Q}(\sqrt{2})$.

A common method for constructing minimal polynomials for numbers that are expressible over $\mathbb{Q}$ is “backwards ”: The number can be set equal to $x$, and the equation can be algebraically manipulated until a monic polynomial in $\mathbb{Q}[x]$ is equal to 0. Finally, if the monic polynomial is not irreducible, then it can be factored into irreducible polynomials $\mathbb{Q}[x]$, and the original number will be a root of one of these. A very example is $\sqrt[4]{2}$:

This method will be further demonstrated with three more examples: One for ${\displaystyle \frac{1+\sqrt{5}}{2}}$, one for $1+{\omega }_5$ where ${\omega }_5$ is a fifth root of unity, and one for $\sqrt[3]{2}+\sqrt[3]{3}$.

Since $x^2-x-1$ is a quadratic and has no roots in $\mathbb{Q}$, it is irreducible over $\mathbb{Q}$. Thus, it is the minimal polynomial over $\mathbb{Q}$ for ${\displaystyle \frac{1+\sqrt{5}}{2}}$.

On the other hand, $x^5-5x^4+10x^3-10x^2+5x-2$ factors over $\mathbb{Q}$ as $(x-2)(x^4-3x^3+4x^2-2x+1)$. Since $1+{\omega }_5$ is not a root of $x-2$, it must be a root of $x^4-3x^3+4x^2-2x+1$. Moreover, this polynomial must be irreducible. This fact can be proven in the following manner: Let $m(x)$ be the minimal polynomial for $1+{\omega }_5$ over $\mathbb{Q}$. Since $\mathbb{Q}(1+{\omega }_5)=\mathbb{Q}({\omega }_5)$, $\mathrm{deg}m(x)=[\mathbb{Q}(1+{\omega }_5):\mathbb{Q}]=[\mathbb{Q}({\omega }_5):\mathbb{Q}]=\phi (5)=4=\mathrm{deg}(x^4-3x^3+4x^2-2x+1)$. (Here $\phi$ denotes the Euler totient function.) Since $m(x)$ divides $x^4-3x^3+4x^2-2x+1$ and they have the same degree, it follows that $m(x)=x^4-3x^3+4x^2-2x+1$.

It turns out that $x^9-15x^6-87x^3-125$ is irreducible over $\mathbb{Q}$. (This can be proven in a manner as above. Note that $[\mathbb{Q}(\sqrt[3]{2}+\sqrt[3]{3}):\mathbb{Q}]=9$.) Thus, it is the minimal polynomial over $\mathbb{Q}$ for $\sqrt[3]{2}+\sqrt[3]{3}$.

Title	examples of minimal polynomials
Canonical name	ExamplesOfMinimalPolynomials
Date of creation	2013-03-22 16:55:18
Last modified on	2013-03-22 16:55:18
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	13
Author	Wkbj79 (1863)
Entry type	Example
Classification	msc 12F05
Classification	msc 12E05
Related topic	IrreduciblePolynomialsObtainedFromBiquadraticFields