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Groups, Rings, and Fields

Groups, rings, and fields are characterized by their operations and their respective inverses. Simply put, if they

\[\displaylines{ \begin{array}{|l|l|} \hline \small{Operations} & \small{Type} \\ \hline +, - & \small{Group}\\ \hline +, -, \times & \small{Ring}\\ \hline +, -, \times, \div & \small{Field}\\ \hline \end{array} }\]

Addition with Additive Inverses

\[\displaylines{ 3-5 = 3 + (-5)\\ \\ 7-3 = 7 + (-3)\\ }\]

Multiplication with Multiplicative Inverses

\[\displaylines{ 10 \div 3 = 10 \times \frac{1}{3}\\ \\ 2 \div 6 = 2 \times \frac{1}{6}\\ }\]

Field Qualification

Consider the following groups:

\[\displaylines{ \begin{array}{lll} \large{\mathbb{Z}} & \text{Integers}\\ \large{\mathbb{R}^{2 \times 3}} & 2 \times 2 \text{ Real Matrices}\\ \large{\mathbb{R}^{2 \times 2}} & 2\times 2 \text{ Real Matrices}\\ \large{\mathbb{Q}} & \text{Rational Numbers}\\ \large{\mathbb{Z / 5 Z}} & \text{Integers } \bmod 5\\ \large{\mathbb{Z / 6 Z}} & \text{Integers } \bmod 6\\ \end{array} }\]

These are a list of qualifying properties for a field

\[\displaylines{ \begin{array}{|l|l|l|l|l|l|} \hline & \mathbb{Z} & \mathbb{R^{2 \times 3}} & \mathbb{R^{2 \times 2}} & \mathbb{Q} & \mathbb{Z / 5Z} & \mathbb{Z / 6Z}\\ \hline \small{\text{Groups Under Addition}} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714}\\ \hline \small{\text{Closed Under Addition}} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714}\\ \hline \small{\text{Additive Inverses}} & \unicode{x2714} & \unicode{x2714}\ & \unicode{x2714} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714}\\ \hline \small{\text{0 Element & Associativity}} & \unicode{x2714} & \unicode{x2714}\ & \unicode{x2714} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714}\\ \hline \small{\text{Commutative under +}} & \unicode{x2714} & \unicode{x2714}\ & \unicode{x2714} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714}\\ \hline \small{\text{Multiplication }} \times & \unicode{x2714} & \color{red}{\unicode{x2718}} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714}\\ \hline \small{\text{Multiplication is commutative}} & \unicode{x2714} & & \color{red}{\unicode{x2718}} & \unicode{x2714} & \unicode{x2714} & \unicode{x2714}\\ \hline \small{\text{Multiplication inverses}} & \color{red}{\unicode{x2718}} & & \color{red}{\unicode{x2718}} & \unicode{x2714} & \unicode{x2714} & \color{red}{\unicode{x2718}}\\ \hline \end{array} }\]

  1. Closed Under Addition means that the sum of any two elements will be in the set.

  2. Additive Inverses means that the negative of each element will be in the set.

  3. 0 Element & Associativity means there’s an additive identity and the associative property holds.

  4. Multiplicative Inverses for non \(0\) numbers. For integers, only \(\{1, -1\}\) have multiplicative inverses (i.e. the multiplicative inverse of 2 is \(\frac{1}{2} \notin \mathbb{Z}\)

    In \(\mathbb{Z / 6Z}\) \(\{2, 3, 4\}\) do not have inverses \(\bmod 6\)

    We can look at \(\mathbb{Z / 6Z}\) and determine that, indeed, all non-\(0\) numbers, are closed under multiplication.

    \[\displaylines{ \begin{array}{|l|l|l|l|l|} \hline & 1 & 2 & 3 & 4\\ \hline 1 & 1 & 2 & 3 & 4\\ \hline 2 & 2 & 4 & 1 & 3\\ \hline 3 & 3 & 1 & 4 & 2\\ \hline 4 & 4 & 3 & 2 & 1\\ \hline \end{array} }\]

  5. Multiplicative Identity (1) The product of a number, and it’s multiplicative inverse is 1 which applies to both \(\displaylines{ \begin{align*} 5 \times \frac{1}{5} &= 1\\ \sqrt[3]{8} \times \frac{1}{\sqrt[3]{8}} &= 1\\ \sqrt[5]{3125} \times \frac{1}{\sqrt[5]{3125}} &= 1\\ \end{align*} }\)

Fields

  • Commutative under \(+\)
  • Both have multiplication \(\times\) \(\Rightarrow\) rings
  • Multiplication is commutative \((a \cdot b = b \cdot a)\) \(\Rightarrow\) commutative rings
  • Multiplicative inverses

Formally, we can say that:

  • A field is a set of element \(F\) with 2 operations \(+\) and \(\times\)

  • \(\bigl \langle F, + \bigr\rangle\) They are commutative under addition

  • \(\bigl \langle F^{\times}, \cdot \bigr\rangle\) They are commutative under multiplication for non-\(0\)

  • Addition and multiplication are linked by the distributive property

    \(a \cdot (b + c) = a \cdot b + a \cdot c\) and \((b + c) \cdot a = b \cdot a + c \cdot a\)

Characteristic of Prime Fields

\(\mathbb{Q}\) or rational numbers are a quotient of 2 integers \(\mathbb{Q} = \{ \frac{a}{b} \text{ where a, b are integers and b} \ne 0 \}\). Rationals are an infinite field

\(\mathbb{Z/5Z}\) and all \(\mathbb{Z}/p\mathbb{Z}\) for any prime number \(p\) are finite fields

If you pick any field \(F\) then it will contain one and only one of the prime fields as a sub-field and we can say that \(F\) is an extension field.

  • If \(F\) is the extension of the integers \(\bmod 2\) then we say “\(F\) has characteristic \(2\)” or \(char(F) = 2\)
  • If \(F\) is the extension of the integers \(\bmod p\) then we say “\(F\) has characteristic \(p\)” or \(char(F) = p\)
  • If \(F\) is the extension of the rational numbers then we say “\(F\) has characteristic \(0\)” or \(char(F) = 0\)
  • There is an infinite number of infinite fields and an infinite number of finite fields.

Types of Fields

  1. Rational Number Fields \(\mathbb{Q}\)
  2. Real Number Fields \(\mathbb{R}\)
  3. Complex Number Fields \(\mathbb{C}\)

Extension Fields for Rational Numbers

There’s an infinite number of extension fields for rational numbers. We can extend rational numbers by adding \(\mathbb{Q}(\sqrt[3]{3})\) or \(\mathbb{Q}(i)\) \((\mathbb{Q}(\sqrt{1}))\)

Let \(\alpha\) be a solution to: \(a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{1}x + a_{0} = 0\) where the coefficients \(a_i\) are fractions which would make \(\mathbb{Q}(\alpha)\) as an extension field of \(\mathbb{Q}\). This is called algebraic extension

Convergent Sequences in calculus are a series of numbers which get closer and closer to some value \(L\) or the limit sequence of \(L\):

\[\displaylines{a_1,a_2,a_3,\cdots \rightarrow L}\]

If you start out with the rational numbers \(Q\) and look at all the convergent sequences and add their limits (\(L\)) to the set of rational numbers you will end up with the real numbers \(\mathbb{R}\) so we can say that real numbers are complete.

Adding \(i = \sqrt{-1}\) to \(\mathbb{R}\) and then adding, subtracting, dividing, and multiplying everything repeatedly you will end up with the complex numbers \(\mathbb{C}\).

\(\mathbb{C}\) is complete, so we cannot extend \(\mathbb{C}\) with new numbers like we did with \(\mathbb{Q}\) by taking limits of sequences or solutions to polynomial equations because any limit sequence of complex numbers will converge to another complex number.