Lord of the Commutative Rings - Numberphile

Key Concepts

• Ring: An algebraic structure with addition and multiplication operations satisfying specific axioms (associativity, identity, inverses for addition, commutativity for commutative rings, distributivity).
• Associativity: The order in which elements are combined doesn't affect the result (e.g., (a + b) + c = a + (b + c)).
• Identity: A special element that, when combined with any other element, leaves the other element unchanged (0 for addition, 1 for multiplication).
• Inverse: An element that, when combined with another element, results in the identity element (additive inverse -a for a).
• Commutativity: The order of elements in an operation doesn't affect the result (a * b = b * a).
• Distributivity: Multiplication distributes over addition (a * (b + c) = a * b + a * c).
• Integers (Z): The set of all whole numbers, including zero and their negatives. A fundamental ring.
• Field: A ring where every nonzero element has a multiplicative inverse.
• Rationals (Q): The set of all numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero. An example of a field.
• Zero Ring: The ring containing only the element zero.
• Even Odd Arithmetic: A ring with two elements, "even" and "odd," with addition and multiplication defined by rules analogous to those of even and odd integers.
• Clock Arithmetic (Z mod nZ): Arithmetic performed on a clock with n hours, where numbers "wrap around" after reaching n.
• Real Numbers (R): The set of all numbers that can be represented on a number line, including rational and irrational numbers.
• Complex Numbers (C): Numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1).
• Gaussian Integers (Z[i]): Complex numbers of the form a + bi, where a and b are integers.
• Gaussian Primes: Prime elements in the ring of Gaussian integers.
• Polynomial Ring (R[x]): The set of all polynomials with coefficients in a ring R.
• Prime Polynomial: A polynomial that cannot be factored into the product of two non-constant polynomials.

What is a Ring?

A ring is an algebraic structure defined by a set, R, equipped with two operations: addition (+) and multiplication (*). These operations must satisfy specific axioms:

• Addition:
  • Associativity: (a + b) + c = a + (b + c) for all a, b, c in R.
  • Identity: There exists an element 0 in R such that a + 0 = a for all a in R.
  • Inverse: For every a in R, there exists an element -a in R such that a + (-a) = 0.
  • Commutativity: a + b = b + a for all a, b in R.
• Multiplication:
  • Associativity: (a * b) * c = a * (b * c) for all a, b, c in R.
  • Identity: There exists an element 1 in R such that a * 1 = a for all a in R.
  • Commutativity (for commutative rings): a * b = b * a for all a, b in R.
• Distributivity: Multiplication distributes over addition: a * (b + c) = a * b + a * c and (b + c) * a = b * a + c * a for all a, b, c in R.

Examples of Rings

Integers (Z)

The integers are a fundamental example of a ring. They satisfy all the ring axioms under ordinary addition and multiplication. The integers are considered the "one ring to rule them all" because any ring contains a copy of the integers (generated by the additive identity 0 and the multiplicative identity 1). Removing elements from the integers, like removing 5, violates the ring axioms because addition and multiplication are no longer well-defined (e.g., 2 + 3 = 5 is no longer possible).

Rationals (Q)

The rationals are a field, which is a special type of ring where every nonzero element has a multiplicative inverse. This means that for any nonzero rational number a, there exists a rational number a⁻¹ such that a * a⁻¹ = 1. The rationals are constructed from the integers by formally adding all possible quotients of integers.

Zero Ring

The zero ring is the simplest ring, containing only the element zero. In this ring, 0 is also the multiplicative identity (0 = 1).

Even Odd Arithmetic

This ring consists of two elements, "even" and "odd," with addition and multiplication defined as follows:

| + | Even | Odd | | :---- | :--- | :--- | | Even | Even | Odd | | Odd | Odd | Even |

| * | Even | Odd | | :---- | :--- | :--- | | Even | Even | Even | | Odd | Even | Odd |

"Even" acts as the additive identity (0), and "odd" acts as the multiplicative identity (1).

Clock Arithmetic (Z mod nZ)

Clock arithmetic involves performing arithmetic operations on a clock with n hours. Numbers "wrap around" after reaching n. For example, in Z mod 12Z (12-hour clock), 13 is equivalent to 1. Even odd arithmetic is equivalent to Z mod 2Z.

Real Numbers (R)

The real numbers are a field consisting of all possible decimal expansions (finite or infinite). They can be constructed from the rationals by "filling in the holes" to include irrational numbers like pi and the square root of 2.

Complex Numbers (C)

The complex numbers are an extension of the real numbers, formed by adding the imaginary unit i (√-1). Every complex number can be written in the form a + bi, where a and b are real numbers.

Gaussian Integers (Z[i])

The Gaussian integers are complex numbers of the form a + bi, where a and b are integers. They form a ring under complex number addition and multiplication. For example, (2 + 3i) * (4 - 2i) = 14 + 8i.

Gaussian Primes

Gaussian integers have unique factorization properties similar to integers. A Gaussian prime is a Gaussian integer that cannot be factored into the product of two non-unit Gaussian integers (units are 1, -1, i, -i). Fermat's theorem characterizes when a prime integer p remains prime in the Gaussian integers: p is prime in Z[i] if and only if p is not congruent to 1 mod 4. For example, 3 is a Gaussian prime, but 5 is not (5 = (2 + i)(2 - i)).

Polynomial Rings (R[x])

A polynomial ring R[x] is the set of all polynomials with coefficients in a ring R. For example, R[x] with R being the real numbers, includes polynomials like 2x⁴ - πx + 3. Polynomials are treated as formal expressions rather than functions. Multiplication and addition of polynomials are performed using the distributive property and the commutativity of multiplication. If the coefficients are from a field (like the real numbers, complex numbers, or rationals), then the polynomial ring has unique factorization.

Importance of Rings

Rings provide a framework for sorting and organizing mathematical objects. Polynomial rings are particularly important because they are versatile tools for modeling functions and describing the world. Calculus, for example, can be viewed as a method for approximating arbitrary functions with polynomials. By imposing relations on polynomial rings, new and interesting rings can be constructed. For example, imposing the relation x² + 1 = 0 on the polynomial ring R[x] results in a ring isomorphic to the complex numbers C, where x plays the role of the imaginary unit i.