Abstract Algebra: Theory and Applications · Problem 21.8.3

Finite Fields — Thomas Judson

Mathematics — Abstract Algebra · minimal-polynomial-linear-algebra-sage

Problem

The subsection Algebraic Elements relies on techniques from linear algebra and contains Theorem 21.1.15: every finite extension is an algebraic extension. This exercise will help you understand this proof. The polynomial r(x)=x4+2x+2r(x) = x^4 + 2x + 2 is irreducible over the rationals (Eisenstein's criterion with prime p=2p = 2). Create a number field that contains a root of r(x)r(x). By Theorem 21.1.15, and the remark following, every element of this finite field extension is an algebraic number, and hence satisfies some polynomial over the base field (it is this polynomial that Sage will produce with the .minpoly() method). This exercise will show how we can use just linear algebra to determine this minimal polynomial. Suppose that a is the generator of the number field you just created with r(x)r(x). Then we will determine the minimal polynomial of t=3a+1t = 3a + 1 using just linear algebra. According to the proof, the first five powers of tt (start counting from zero) will be linearly dependent. (Why?) So a nontrivial relation of linear dependence on these powers will provide the coefficients of a polynomial with tt as a root. Compute these five powers, then construct the correct linear system to determine the coefficients of the minimal polynomial, solve the system, and suitably interpret its solutions. Hints: The vector() and matrix() commands will create vectors and matrices, and the .solve\_right() method for matrices can be used to find solutions. Given an element of the number field, which will necessarily be a polynomial in the generator a, the .vector() method of the element will provide the coefficients of this polynomial in a list.
Intuitive explanation

The key insight is that a finite extension has finite dimension as a vector space over the base field, so any n+1n+1 elements must be linearly dependent. Here, the five powers t0,t1,t2,t3,t4t^0, t^1, t^2, t^3, t^4 live in a 4-dimensional vector space (since [Q(a):Q]=4[ℚ(a):ℚ] = 4), guaranteeing a nontrivial linear dependence relation. By expressing each power as a vector of coefficients in the basis {1,a,a2,a3}\{1, a, a^2, a^3\} and solving for the dependence relation, you'll find the minimal polynomial of tt directly from linear algebra—no polynomial division needed.

Intuitive explanation

The key insight is that a finite extension has finite dimension as a vector space over the base field, so any n+1n+1 elements must be linearly dependent. Here, the five powers t0,t1,t2,t3,t4t^0, t^1, t^2, t^3, t^4 live in a 4-dimensional vector space (since [Q(a):Q]=4[ℚ(a):ℚ] = 4), guaranteeing a nontrivial linear dependence relation. By expressing each power as a vector of coefficients in the basis {1,a,a2,a3}\{1, a, a^2, a^3\} and solving for the dependence relation, you'll find the minimal polynomial of tt directly from linear algebra—no polynomial division needed.

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