2.1: Fields
In this section, we'll seek to answer the questions:
- What are binary operations?
- What is a field? What sorts of things can one do in a field?
- What are examples of fields?
We now begin the process of abstraction. We will do this in stages, beginning with the concept of a field . First, we need to formally define some familiar sets of numbers.
The rational numbers , denoted by \(\mathbb{Q}\text{,}\) is the set
\begin{equation*} \mathbb{Q} = \left \{\dfrac{a}{b}: a,b\in \mathbb{Z}, \ b\ne 0\right \}\text{.} \end{equation*}
Recall that in elementary school, you learned that two fractions \(\dfrac{a}{b}, \dfrac{c}{d} \in \mathbb{Q}\) are equivalent if and only if \(ad=bc\text{.}\)
Prove that our elementary school definition of equivalent fractions is an equivalence relation. Recall Definition: Rational Numbers.
We likely have an intuitive idea of what is meant by \(\mathbb{R}\text{,}\) the set of real numbers. Defining \(\mathbb{R}\) rigorously is actually quite difficult, and occupies a significant amount of time in a first course in real analysis. Thus, we will make use of your intuition.
Out of \(\mathbb{R}\) we may build the complex numbers.
The complex numbers consist of all expressions of the form \(a+bi\text{,}\) where \(a,b\in \mathbb{R}\) and \(i^2 = -1\text{.}\) Given \(z = a+bi\text{,}\) we say \(a\) is the real part of \(z\) and \(b\) is the imaginary part . The set of complex numbers is denoted \(\mathbb{C}\text{.}\)
As was mentioned in the Introduction, algebra comes from an Arabic word meaning “the reunion of broken parts”. We therefore need a way of combining two elements of a set into one; we turn to a particular type of function, known as a binary operation, to accomplish this.
Let \(X\) be a nonempty set. A function \(\star : X \times X \to X\) is called a binary operation . If \(\star\) is a binary operation on \(X\text{,}\) we say that \(X\) is closed under the operation \(\star\) . [Given \(a,b\in X\text{,}\) we usually write \(a\star b\) in place of the typical function notation, \(\star(a,b)\text{.}\)]
Which of \(+, -, \cdot, \div\) are binary operations:
- on \(\mathbb{R}\text{?}\)
- on \(\mathbb{Q}\text{?}\)
- on \(\mathbb{Z}\text{?}\)
- on \(\mathbb{N}\text{?}\)
- on \(\mathbb{C}\text{?}\) (Recall that for \(a_1 + b_1 i, a_2 + b_2 i \in \mathbb{C}\text{,}\) \((a_1 + b_1 i) + (a_2 + b_2 i) := (a_1 + a_2) + (b_1 + b_2)i\) and \((a_1 + b_1 i)(a_2 + b_2 i) := (a_1 a_2 - b_1 b_2) + (a_1 b_2 + b_1 a_2) i\text{.}\))
Choose your favorite nonempty set \(X\) and describe a binary operation different than those in Investigation 2.1.1 .
The hallmark of modern pure mathematics is the use of axioms . An axiom is essentially an unproved assertion of truth. Our use of axioms serves several purposes.
From a logical perspective, axioms help us avoid the problem of infinite regression (e.g., asking How do you know? over and over again). That is, axioms give us very clear starting points from which to make our deductions.
To that end, our first abstract algebraic structure captures and axiomatizes familiar behavior about how numbers can be combined to produce other numbers of the same type.
A field is a nonempty set \(F\) with at least two elements and binary operations \(+\) and \(\cdot\text{,}\) denoted \((F,+,\cdot)\text{,}\) and satisfying the following field axioms :
- Given any \(a,b,c\in F\text{,}\) \((a+b)+c = a+(b+c)\text{.}\) (Associativity of addition)
- Given any \(a,b\in F\text{,}\) \(a+b= b+a\text{.}\) (Commutativity of addition)
- There exists an element \(0_F\in F\) such that for all \(a\in F\text{,}\) \(a+0_F = 0_F + a = a\text{.}\) (Additive identity)
- Given any \(a\in F\) there exists a \(b\in F\) such that \(a+b = b + a =0_F\text{.}\) (Additive inverse)
- Given any \(a,b,c\in F\text{,}\) \((a\cdot b)\cdot c = a\cdot (b\cdot c)\text{.}\) (Associativity of multiplication)
- Given any \(a,b\in F\text{,}\) \(a\cdot b = b\cdot a\text{.}\) (Commutativity of multiplication)
- There exists an element \(1_F\in F\) such that for all \(a\in F\text{,}\) \(1_F\cdot a = a\cdot 1_F = a\text{.}\) (Multiplicative identity)
- For all \(a\in F\text{,}\) \(a\ne 0_F\text{,}\) there exists a \(b\in F\) such that \(a\cdot b = b\cdot a = 1_F\text{.}\) (Multiplicative inverse)
- For all \(a,b,c\in F\text{,}\) \(a\cdot (b+c) = a\cdot b + a\cdot c\text{.}\) (Distributive property I)
- For all \(a,b,c\in F\text{,}\) \((a+b)\cdot c = a\cdot c + b\cdot c\text{.}\) (Distributive property II)
We will usually write \(a\cdot b\) as \(ab\text{.}\) Additionally, we will usually drop the subscripts on \(0,1\) unless we need to distinguish between fundamentally different identities in different fields.
Which of the following are fields under the specified operations? For most, a short justification or counterexample is sufficient.
- \(\mathbb{N}\) under the usual addition and multiplication operations
- \(\mathbb{Z}\) under the usual addition and multiplication operations
- \(2\mathbb{Z}\text{,}\) the set of even integers, under the usual addition and multiplication operations
- \(\mathbb{Q}\) under the usual addition and multiplication operations
- \(\mathbb{Z}_{6}\) under addition and multiplication modulo 6
- \(\mathbb{Z}_{5}\) under addition and multiplication modulo 5
- \(\mathbb{R}\) under the usual addition and multiplication operations
- \(\mathbb{C}\) under the complex addition and multiplication defined in Investigation 2.1.1
- \(\mathcal{M}_2(\mathbb{R}) := \left \{\left(\begin{matrix}a & b \\ c & d \end{matrix} \right) : a,b,c,d\in\mathbb{R}\right \}\) 1 , the set of \(2\times 2\) matrices with real coefficients using the usual definition of matrix multiplication 2 and matrix addition.
- 1
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For students who have taken a linear algebra course.
- 2
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Recall that, if \(\left(\begin{matrix}a_1 & b_1 \\ c_1 & d_1 \end{matrix} \right), \left(\begin{matrix}a_2 & b_2 \\ c_2 & d_2 \end{matrix} \right)\in \mathcal{M}_2(\mathbb{R})\text{,}\) then \[\left(\begin{matrix}a_1 & b_1 \\ c_1 & d_1 \end{matrix} \right) \cdot \left(\begin{matrix}a_2 & b_2 \\ c_2 & d_2 \end{matrix} \right) = \left(\begin{matrix}a_1 a_2 + b_1 c_2 & a_1 b_2 + b_1 d_2 \\ c_1 a_2 + d_1 c_2 & c_1 b_2 + d_1 d_2 \end{matrix} \right)\text{.} \nonumber\]
In the Investigation 2.1.2 , you determined which of sets of familiar mathematical objects are and are not fields. Notice that you have been working with fields for years and that our abstraction of language to that of fields is simply to allow us to explore the common features at the same time - it is inefficient to prove the same statement about every single field when we can prove it once and for all about fields in general.
Let \(F\) be a field.
- The additive identity \(0\) is unique.
- For all \(a\in F\text{,}\) \(a \cdot 0 = 0\cdot a = 0\text{.}\)
- Additive inverses are unique.
- The multiplicative identity \(1\) is unique.
- Multiplicative inverses are unique.
- \((-1)\cdot (-1) = 1\)
- Hint
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Note that we are saying that the additive inverse of the multiplicative identity times itself equals the multiplicative identity. You should use only the field axioms and the properties previously established in this theorem.
One consequence of Theorem 2.1.1 is that, given \(a\in F\text{,}\) \(b\in F\setminus \{0\}\text{,}\) we may refer to \(-a\) as the additive inverse of \(a\text{,}\) and \(b^{-1}\) as the multiplicative inverse of \(b\text{.}\) We will thus employ this familiar terminology henceforth.
For which \(n > 1\) is \(\mathbb{Z}_n\) a field? Compute some examples, form a conjecture, and prove your conjecture.