Field Theory and Polynomials

Field theory is the study of fields and their extensions. It provides the necessary framework for understanding the roots of polynomials, geometric constructions, and the solvability of equations. In this lesson, we explore the structure of polynomial rings and the mechanics of extending a field to include roots of irreducible polynomials.

Polynomial Rings $F[x]$

Let $F$ be a field. The set of all polynomials with coefficients in $F$, denoted $F[x]$, forms a Commutative Ring. Specifically, $F[x]$ is a Euclidean Domain, meaning we can perform division with remainders.

The Division Algorithm

For any $f(x),g(x)\in F[x]$ with $g(x)\ne 0$, there exist unique $q(x),r(x)\in F[x]$ such that: $f(x)=g(x)q(x)+r(x)$ where either $r(x)=0$ or $\mathrm{deg}(r)<\mathrm{deg}(g)$.

Irreducibility

A polynomial $p(x)\in F[x]$ of degree $\ge 1$ is irreducible if it cannot be written as the product of two non-constant polynomials.

• Eisenstein’s Criterion: A tool for identifying irreducibility over $\mathbb{Q}$. If for a prime $p$, $p$ divides all coefficients $a_i$ except the leading $a_n$, and $p^2$ does not divide $a_0$, then the polynomial is irreducible.

Field Extensions

A field $L$ is an extension of a field $K$ (written $L/K$) if $K\subseteq L$. We can view $L$ as a vector space over the “base field” $K$.

• The degree of the extension, $[L:K]$, is the dimension of $L$ as a vector space over $K$.

Simple Extensions $K(\alpha )$

If $\alpha \in L$, then $K(\alpha )$ is the smallest subfield of $L$ containing both $K$ and $\alpha$.

• If $\alpha$ is a root of an irreducible polynomial $p(x)\in K[x]$, then $K(\alpha )\cong K[x]/⟨p(x)⟩$.
• In this case, $\alpha$ is algebraic over $K$, and $p(x)$ is its minimal polynomial.

Algebraic and Transcendental Elements

An element $\alpha$ in an extension $L/K$ is:

1. Algebraic if there exists a non-zero $f(x)\in K[x]$ such that $f(\alpha )=0$.
2. Transcendental if no such polynomial exists (e.g., $\pi$ and $e$ over $\mathbb{Q}$).

Algebraic Extensions

An extension $L/K$ is algebraic if every element in $L$ is algebraic over $K$.

• Theorem: Every finite extension ($[L:K]<\infty$) is an algebraic extension. The converse is not true (e.g., the field of all algebraic numbers $\bar{\mathbb{Q}}$ is algebraic over $\mathbb{Q}$ but has infinite degree).

Splitting Fields

A splitting field of a polynomial $f(x)\in K[x]$ is the smallest extension $L$ of $K$ in which $f(x)$ factors completely into linear factors: $f(x)=c(x-{\alpha }_1)(x-{\alpha }_2)\dots (x-{\alpha }_n)$ Splitting fields are unique up to isomorphism. They are essential for Galois Theory, as they allow us to study the symmetry of the roots.

Finite Fields (Galois Fields)

Finite fields have applications in cryptography and coding theory. A finite field must have $p^n$ elements for some prime $p$ and integer $n\ge 1$.

• Denoted ${\mathbb{F}}_{p^n}$ or $GF(p^n)$.
• The multiplicative group of a finite field is always cyclic.
• Construction: To create ${\mathbb{F}}_4$, take ${\mathbb{F}}_2=\{0,1\}$ and quotient by an irreducible quadratic like $x^2+x+1$. The elements become $\{0,1,x,x+1\}$.

Practical Example: Polynomial Division in Python

We can represent polynomials as lists of coefficients (lowest degree first).

def poly_div(f, g):
    """Simple polynomial division over Q represented as coefficient lists."""
    res = [0] * max(0, len(f) - len(g) + 1)
    rem = list(f)
    for i in range(len(res) - 1, -1, -1):
        res[i] = rem[i + len(g) - 1] / g[-1]
        for j in range(len(g)):
            rem[i + j] -= res[i] * g[j]
    
    # Clean up trailing zeros
    while rem and abs(rem[-1]) < 1e-9: rem.pop()
    return res, rem

# (x^2 - 1) / (x - 1)
f = [-1, 0, 1] 
g = [-1, 1]
q, r = poly_div(f, g)
print(f"Quotient: {q}, Remainder: {r}") # Expect [1, 1], []

Significance of Field Theory

Field theory solved several ancient geometric puzzles, such as the impossibility of “squaring the circle” or “trisecting the angle” using only compass and straightedge. These geometric operations correspond to constructing fields of degree $2^k$; since $\pi$ is transcendental and $\sqrt[3]{2}$ involves a degree 3 extension, they are algebraically impossible.

Exercise