Problem 2 - A Course in Galois Theory

Question

Suppose that \(K(\alpha): K\) is a finite simple extension. For each \(\beta\) in \(K(\alpha)\), let \(T_{\alpha}(\beta)=\alpha \beta . T_{\alpha}\) is a linear mapping of \(K(\alpha)\) (considered as a vector space over \(K\) ) into itself. Show that \(\operatorname{det}\left(x I-T_{\alpha}\right)\) is the minimal polynomial of \(\alpha\) over \(K\).

Step-by-Step Solution

Answer

The determinant \(\operatorname{det}(xI-T_{\alpha})\) is the minimal polynomial of \(\alpha\) over \(K\).

Step 1: Understand the Linear Transformation

First, we need to understand the linear mapping \(T_{\alpha}(\beta) = \alpha \beta\) for \(\beta \) in \(K(\alpha)\). This mapping involves multiplying any element of the extension \(K(\alpha)\) by \(\alpha\). Since \(K(\alpha)\) is a vector space over \(K\) of degree \(n\), we consider an ordered basis \(\{1, \alpha, \alpha^2, \ldots, \alpha^{n-1}\}\).

Step 2: Construct the Matrix Representation

Considering the basis \(\{1, \alpha, \alpha^2, \ldots, \alpha^{n-1}\}\), the mapping \(T_{\alpha}\) represented as a matrix relative to this basis involves shifting each \(\alpha^i\) to \(\alpha^{i+1}\) and applying the relation \(\alpha^n = c_0 + c_1 \alpha + \cdots + c_{n-1} \alpha^{n-1} \) (where \(c_i\) are in \(K\) and constitute the minimal polynomial's coefficients). This matrix is called the companion matrix of the minimal polynomial.

Step 3: Calculate the Characteristic Polynomial

The characteristic polynomial of the matrix \(T_{\alpha}\) is given by \(\det(xI - T_{\alpha})\), where \(I\) is the identity matrix. The companion matrix being used ensures this characteristic polynomial is identical to the minimal polynomial of \(\alpha\) over \(K\) which is \(x^n - c_{n-1} x^{n-1} - \cdots - c_0\).

Step 4: Conclude with the Minimal Polynomial

Finally, according to properties of companion matrices, this characteristic polynomial (from Step 3) is indeed the minimal polynomial of \(\alpha\) over \(K\). Thus, \(\operatorname{det}(xI-T_{\alpha})\) is the minimal polynomial we sought.

Key Concepts

Finite Simple Extension

A finite simple extension of a field \( K \) is an extension field created by adding a single element, called a "primitive element." In this context, we denote this element as \( \alpha \). Think of it as extending \( K \) by the simplest means possible, via adding just one new element. This results in a field \( K(\alpha) \) that contains \( K \) and \( \alpha \). All elements of this new field can be expressed as a linear combination of powers of \( \alpha \).

A critical point to understand is the concept of the degree of the extension. The degree is essentially the dimension of the new field as a vector space over the original field \( K \), denoted as \( [K(\alpha) : K] = n \). This means \( \alpha \) satisfies a polynomial equation with coefficients in \( K \) of degree \( n \).

The extension is referred to as "finite" because it adds only a finite number of new elements, and "simple" because it boils down to just one element: \( \alpha \).

Linear Mapping

In the context of fields and vector spaces, a linear mapping or linear transformation is a mapping that respects the operations of addition and scalar multiplication. For our problem, we consider a particular linear mapping, \( T_{\alpha}(\beta) = \alpha\beta \), where \( \alpha \) is the fixed element from the finite simple extension, and \( \beta \) is an element from \( K(\alpha) \).

This transformation can be visualized as multiplying each element of the vector space (in this case, expressions in terms of \( \alpha \)) by \( \alpha \) itself. It's crucial to recognize that this action results in a new element, still in the same vector space, following the structure of \( K(\alpha) \).

• Respects addition: \( T_{\alpha}(\beta_1 + \beta_2) = \alpha(\beta_1 + \beta_2) = \alpha\beta_1 + \alpha\beta_2 \)
• Respects scalar multiplication: \( T_{\alpha}(c\beta) = \alpha(c\beta) = c(\alpha\beta) \), with \( c \) in \( K \).

Minimal Polynomial

In field theory, the minimal polynomial of an algebraic element \( \alpha \) over a base field \( K \) is the monic polynomial of lowest degree such that \( \alpha \) is a root. This polynomial captures the essence of how \( \alpha \) satisfies algebraic relations over \( K \).

The minimal polynomial is unique and irreducible over \( K \), which means it cannot be factored into lower-degree polynomials with coefficients in \( K \). Its role is crucial because it defines the algebraic structure of the extension \( K(\alpha) \) by linking \( \alpha \) back to \( K \).

In our problem, the determination of the minimal polynomial is achieved via the characteristic polynomial of a linear map \( T_{\alpha} \), realized through its matrix form. The determinant of \( xI - T_{\alpha} \) ends up being the minimal polynomial, as the companion matrix structure ensures the characteristic polynomial aligns with the minimal polynomial.

Companion Matrix

A companion matrix is a specific type of matrix used to encode a polynomial, especially useful in finite field extensions and Galois Theory. Given a monic polynomial of degree \( n \), the companion matrix is an \( n \times n \) matrix that shifts polynomial coefficients to encapsulate the action of a linear transformation associated with simple extensions.

In our specific problem, when you have the polynomial representing the minimal polynomial of \( \alpha \), the companion matrix effectively governs the linear map \( T_{\alpha} \). For a minimal polynomial \( x^n + a_{n-1}x^{n-1} + \cdots + a_0 \), the companion matrix \( C \) looks like:\[C = \begin{bmatrix}0 & 0 & \cdots & 0 & -a_0 \1 & 0 & \cdots & 0 & -a_1 \0 & 1 & \cdots & 0 & -a_2 \\vdots & \vdots & \ddots & \vdots & \vdots \0 & 0 & \cdots & 1 & -a_{n-1} \\end{bmatrix}\]

This structure of the companion matrix makes it straightforward to compute the characteristic polynomial, demonstrating directly that this polynomial is indeed the minimal polynomial of \( \alpha \). Thus, the companion matrix serves as a convenient tool for linking endomorphisms in finite extensions to their defining algebraic polynomials.