Galois Theory

Galois theory provides a framework for reducing complex problems concerning field extensions into the realm of group theory, making them significantly easier to solve and understand.

The Galois Group

In modern algebra, the Galois group is defined in relation to a field extension $L/K$. It is specifically the group comprising all automorphisms of the larger field $L$ that completely fix every element within the base field $K$. This means the automorphisms are functions that map $L$ to itself while leaving the underlying structure of $K$ entirely undisturbed.

Historically, this was viewed as the group of permutations among the roots of a polynomial, given that any algebraic equation satisfied by those roots remains true even after the roots are permuted.

def check_automorphism(elements, op_add, op_mult, mapping):
  for a in elements:
      for b in elements:
          if mapping(op_add(a, b)) != op_add(mapping(a), mapping(b)): return False
          if mapping(op_mult(a, b)) != op_mult(mapping(a), mapping(b)): return False
  return True

# Complex conjugation in C fixing R
mapping = lambda z: z.conjugate()
elements = [1+1j, 2-3j, 0+0j, 1+0j]
print(f"Is conjugation an automorphism? {check_automorphism(elements, lambda a,b: a+b, lambda a,b: a*b, mapping)}")

Fundamental Theorem of Galois Theory

The centerpiece of this subject is the Fundamental Theorem of Galois Theory. This theorem establishes a formal, direct correspondence between field theory and group theory.

Specifically, it proves that for a given field extension, there is a distinct relationship between its intermediate subfields and the subgroups of its overall Galois group. This one-to-one mapping empowers mathematicians to directly analyze the traits of field extensions by studying the properties of their corresponding groups.

Solvability by Radicals

Galois theory is famously used to determine whether a polynomial equation is solvable by radicals. An equation holds this property if its roots can be written out using a formula consisting solely of integers, the four primary arithmetic operations (addition, subtraction, multiplication, and division), and $n$-th roots (radicals).

The theory definitively shows that a polynomial equation is solvable by radicals if and only if its corresponding Galois group is categorized as a “solvable group.” This explains why generic polynomials of degree five or higher lack a universal root formula.

def is_solvable_S5():
  # Symmetric group S5 is not solvable because its 
  # only non-trivial normal subgroup is A5, which is simple.
  print("Is S5 solvable? False")
  print("Wait... can we check S3?")
  # S3 -> A3 -> {e} is a subnormal series with Abelian factors.
  print("Is S3 solvable? True")

is_solvable_S5()

References & Further Reading

• Wikipedia: Galois theory (CC-BY-SA 3.0)