Let αbe an algebraic number and f, g ∈ℚ{x} of degrees n, m ∈ℕ2 1 such that f is the minimal poly

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Let αbe an algebraic number and f, g ∈ℚ{x} of degrees n, m...

Let $\alpha$ be an algebraic number and $f, g \in \mathbb{Q}\{x\}$ of degrees $n, m \in \mathbb{N}_{2} 1$ such that $f$ is the minimal polynomial of $\alpha$. We want to compute the minimal polynomial of $g(\alpha)$ and therefore may assume that $n>m$.

Key Concepts

Resultants and Elimination Techniques

Resultants provide a systematic method for eliminating variables between polynomials to find common roots. In the context of finding the minimal polynomial of g(?), elimination techniques like computing the resultant of the minimal polynomial of ? and an equation derived from g can be used to derive a polynomial that g(?) satisfies.

Polynomial Degree

The degree of a polynomial, defined as the highest power of the variable with a non-zero coefficient, is an important property that influences the behavior of the polynomial, including the number of possible roots and the structure of its field extension when the polynomial is irreducible over a given field.

Polynomial Function Composition and Transformation

Composing or evaluating one polynomial at the output of another, such as computing g(?) when ? is algebraic and f is its minimal polynomial, involves understanding how algebraic dependencies transform under polynomial maps. This concept is crucial in determining the minimal polynomial of the transformed value.

Minimal Polynomial

The minimal polynomial of an algebraic number is the unique monic polynomial with rational coefficients of lowest degree that has the number as a root. It encapsulates the essential algebraic properties of the number and determines the degree of the corresponding field extension over the rationals.

Algebraic Numbers

An algebraic number is a number that satisfies a non-zero polynomial equation with rational coefficients. This concept is foundational in number theory and field theory, as it leads to the construction of algebraic extensions of the rationals and provides a framework for analyzing solutions of polynomial equations.

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