Rings, Fields, and Integral Domains

While group theory analyzes systems with a single operation, ring theory deals with sets that have two interconnected binary operations, typically identified as addition ($+$) and multiplication ($\cdot$).

Defining a Ring

A ring is a set $R$ equipped with these two operations that must adhere to three specific layers of axioms:

1. Addition forms an Abelian group: Under the operation of addition, the set must be associative and commutative. It must contain an additive identity ($0$), and every element $a$ must have an additive inverse ($-a$).
2. Multiplication forms a Monoid: Under multiplication, the set must be associative. Furthermore, it must possess a multiplicative identity ($1$) such that multiplying any element by $1$ leaves it unchanged.
3. Distributivity: Multiplication must distribute over addition from both the left and the right. This means $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$, and similarly $(b + c) \cdot a = (b \cdot a) + (c \cdot a)$.

def is_ring_add_abelian(elements, add_op, zero):
  for a in elements:
      if add_op(a, zero) != a: return False
      for b in elements:
          if add_op(a, b) != add_op(b, a): return False
          if add_op(a, b) not in elements: return False
  return True

# Z/4Z Addition
elements = [0, 1, 2, 3]
add = lambda a, b: (a + b) % 4
print(f"Is addition in Z/4Z Abelian? {is_ring_add_abelian(elements, add, 0)}")

Integral Domains

A domain is defined as a nonzero ring that lacks nonzero zero-divisors—meaning you cannot multiply two nonzero elements together and get zero as a result.

Building upon this, an Integral Domain is simply a domain that is also commutative. Thus, it is a commutative, nonzero ring where the product of any two nonzero elements is guaranteed to be nonzero. This property is what allows basic cancellation laws to function in algebraic equations.

def find_zero_divisors(elements, mult_op, zero):
  divisors = []
  for a in elements:
      if a == zero: continue
      for b in elements:
          if b == zero: continue
          if mult_op(a, b) == zero:
              divisors.append(a)
              break
  return divisors

# Z/6Z multiplication
mult = lambda a, b: (a * b) % 6
z6 = [0,1,2,3,4,5]
print(f"Zero divisors in Z/6Z: {find_zero_divisors(z6, mult, 0)}")

Principal Ideal Domains (PID)

In ring theory, an ideal is a subset of a ring that absorbs multiplication by any ring element. An ideal is considered “principal” if it can be entirely generated by multiples of a single element from the ring.

A Principal Ideal Domain (PID) is defined strictly as an integral domain wherein every single ideal is a principal ideal. The set of integers ($\mathbb{Z}$) is the classic example of a PID.

References & Further Reading

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

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)

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)

Advanced Group Theory

Advanced group theory relies heavily on understanding how subgroups partition a group and how these partitions can form new mathematical structures.

Normal Subgroups

When working with a subgroup $N$ inside a larger group $G$, we can form subsets called cosets. For any element $g$ in $G$, the left coset is the set formed by multiplying $g$ with every element in $N$ ($gN$). The right coset is formed by multiplying in the opposite order ($Ng$).

A normal subgroup is a special type of subgroup where it is invariant under conjugation. Formally, this means that $gng^{-1}$ is always an element of $N$ for any $n$ in $N$ and any $g$ in $G$. An equivalent and highly useful way to define a normal subgroup is that its left and right cosets are identical: $gN = Ng$ for every element $g$ in the group.

def is_normal(G, H, op):
  for g in G:
      left_coset = set([op(g, h) for h in H])
      right_coset = set([op(h, g) for h in H])
      if left_coset != right_coset:
          return False
  return True

# In Abelian groups, all subgroups are normal
G = list(range(12))
H = [0, 4, 8]
add_mod_12 = lambda a, b: (a + b) % 12
print(f"Is {H} normal in Z/12Z? {is_normal(G, H, add_mod_12)}")

Quotient Groups

The primary reason normal subgroups are so important is that they allow the construction of quotient groups (or factor groups).

If $N$ is a normal subgroup of $G$, the quotient group, denoted as $G/N$, is defined as the set of all left cosets of $N$ inside $G$. The elements of this new group are entire sets (the cosets). The group operation is defined by multiplying the representatives of the cosets: $(aN)(bN) = (ab)N$. This operation is only mathematically valid and well-defined when $N$ is normal.

Lagrange’s Theorem

Lagrange’s Theorem describes a fundamental limitation on the size of subgroups within finite groups. The theorem states that if a group $G$ has a finite number of elements (its order, denoted $|G|$), then the order of any subgroup $H$ must perfectly divide the order of $G$.

The number of distinct cosets of $H$ in $G$ is called the index of $H$, written as $[G : H]$. Lagrange’s theorem proves that the total size of the group is the product of the subgroup’s size and its index: $|G| = [G : H] \cdot |H|$

def get_cosets(G, H, op):
  cosets = set()
  for g in G:
      # Left coset gH = {g * h for h in H}
      coset = frozenset([op(g, h) for h in H])
      cosets.add(coset)
  return cosets

cosets = get_cosets(G, H, add_mod_12)
print(f"Number of cosets (index): {len(cosets)}")
print(f"Subgroup size: {len(H)}")
print(f"Group size: {len(G)}")
print(f"Does |H| divide |G|? {len(G) % len(H) == 0}")

References & Further Reading

• Wikipedia: Normal subgroup (CC-BY-SA 3.0)
• Wikipedia: Group (mathematics) (CC-BY-SA 3.0)

Group Theory Fundamentals

In mathematics, a group is a foundational structure used to study symmetry and the behavior of operations.

Formal Definition of a Group

As defined by standard algebra, a group consists of an ordered pair $(G, \cdot)$. This pairs a set of elements, $G$, with a binary operation, typically denoted as "$\cdot$", which combines any two elements to produce a third.

For this pairing to qualify as a group, it must strictly satisfy three axioms:

1. Associativity: Grouping does not affect the outcome. For any elements $a, b,$ and $c$, the equation $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ must hold true.
2. Identity Element: The set must contain a unique element $e$, such that combining it with any element $a$ leaves $a$ unchanged: $e \cdot a = a$ and $a \cdot e = a$.
3. Inverse Element: Every element $a$ must have a corresponding inverse element, often written as $a^{-1}$. Combining an element with its inverse always yields the identity element: $a \cdot a^{-1} = e$ and $a^{-1} \cdot a = e$.

Additionally, the operation must be closed over the set, meaning the result of $a \cdot b$ is always an element within $G$. If the operation is also commutative ($a \cdot b = b \cdot a$), the group is known as an Abelian group.

def check_group_axioms(elements, op, identity):
  # 1. Closure (Simplified for finite set)
  for a in elements:
      for b in elements:
          if op(a, b) not in elements: return False, "Closure failed"
  
  # 2. Identity
  for a in elements:
      if op(a, identity) != a or op(identity, a) != a:
          return False, f"Identity failed for {a}"
  
  # 3. Inverses
  for a in elements:
      found_inverse = False
      for b in elements:
          if op(a, b) == identity and op(b, a) == identity:
              found_inverse = True
              break
      if not found_inverse: return False, f"Inverse failed for {a}"
      
  return True, "All axioms satisfied"

# Test Z/4Z under addition
z4 = [0, 1, 2, 3]
add_mod_4 = lambda a, b: (a + b) % 4
print(f"Z/4Z result: {check_group_axioms(z4, add_mod_4, 0)}")

# Test Z/4Z under multiplication (Identity = 1)
mult_mod_4 = lambda a, b: (a * b) % 4
print(f"Z/4Z mult result: {check_group_axioms(z4, mult_mod_4, 1)}")

Subgroups and Isomorphisms

A subset $H$ of a group $G$ that independently forms a group under the same operation is called a subgroup. To verify if $H$ is a subgroup, one can check if $g^{-1} \cdot h$ remains in $H$ for all elements $g, h$ in $H$.

When analyzing the relationship between two groups, we use a mapping called a homomorphism. This function, $\varphi: G \to H$, preserves the structural integrity of the operations such that $\varphi(a \cdot b) = \varphi(a) * \varphi(b)$. If this mapping is bijective (a perfect one-to-one correspondence), it is called an isomorphism, indicating the two groups are structurally identical.

References & Further Reading

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