Field Theory and Extensions

A field is one of the most structurally complete objects in algebra, guaranteeing that all basic arithmetic operations can be performed reliably.

The Definition of a Field

A field is a set $F$ that possesses two binary operations: addition ($+$) and multiplication ($\cdot$). For a set to be considered a field, these operations must satisfy a rigorous set of axioms:

• Both addition and multiplication must be associative and commutative.
• The set must contain distinct identities for addition ($0$) and multiplication ($1$).
• Multiplication must distribute over addition.
• Every element must have an additive inverse. Crucially, every non-zero element must also have a multiplicative inverse (which allows for division).

In simpler terms, a field is a commutative ring where $0 \neq 1$, and division by any non-zero element is always possible.

Characteristic of a Field

The characteristic of a field identifies the smallest positive integer $n$ required such that adding the multiplicative identity ($1$) to itself $n$ times results in $0$ ($n \cdot 1 = 0$).

If such an integer exists, it is mathematically proven that it must be a prime number $p$, and the field is said to have characteristic $p$. Conversely, if repeated addition of $1$ never reaches $0$, the field is defined as having a characteristic of $0$ (for example, the field of rational numbers $\mathbb{Q}$).

def get_characteristic(identity_mult, add_op, zero):
  current = identity_mult
  for p in range(1, 1000):
      if current == zero:
          return p
      current = add_op(current, identity_mult)
  return 0

# Test Z/5Z
print(f"Characteristic of Z/5Z: {get_characteristic(1, lambda a, b: (a+b)%5, 0)}")
print(f"Characteristic of Z/997Z: {get_characteristic(1, lambda a, b: (a+b)%997, 0)}")

Field Extensions

A field extension, commonly written as $F/E$, occurs when a smaller field $E$ acts as a subfield within a larger field $F$. Because $E$ is closed under its operations, the larger field $F$ can be analyzed as a vector space built over the scalars provided by $E$. The dimension of this vector space is referred to as the degree of the extension.

Algebraic and Transcendental Elements

When examining elements in an extension field $F$ relative to the base field $E$, they fall into two categories:

• Algebraic: An element $x$ is algebraic if it serves as the root of a non-zero polynomial constructed entirely using coefficients from the base field $E$.
• Transcendental: If an element is not algebraic—meaning no such polynomial can possibly exist to evaluate to zero with $x$ as its root—it is called a transcendental element.

def evaluate_poly(coeffs, x, mod=None):
  res = 0
  for i, c in enumerate(coeffs):
      term = c * (x**i)
      res += term
  return res if mod is None else res % mod

# Is sqrt(2) algebraic over Q? 
# Minimal polynomial: x^2 - 2
coeffs = [-2, 0, 1] 
import math
sqrt2 = math.sqrt(2)
print(f"p(sqrt(2)) = {evaluate_poly(coeffs, sqrt2)}")

References & Further Reading

• Wikipedia: Field (mathematics) (CC-BY-SA 3.0)