0 – Notation and definitions: fields and polynomials

(This is part of my series Field theory for high-schoolers.)

In this first post, we define the central concepts of fields and field extensions; and we define polynomials, which will be our main tool in studying field extensions. Those concepts are probably already familiar to the reader, but we introduce them for the sake of completeness, and to present some notation.

0.1 Fields

A field is a set of elements together with two operations, that behave in the same way as the rational, real, or complex numbers with the operations of addition and multiplication: numbers can be added, substracted, multiplied, and divided by a non-zero number, and all those operations obey the usual rules. The formal definition is as follows, but feel free to skip it. The important fact for us is that $\mathbf{R}$ and $\mathbf{C}$ are fields.

A field is a set $F$ with two binary operations called addition and multiplication. The operation of addition is denoted with the symbol $+$, and the operation of multiplication is denoted with the symbol $\times$ or $\cdot$, or simply by juxtaposition if no ambiguity ensues. The operations satisfy the following:

  • $F$ is closed under addition and multiplication, meaning that addition or multiplication of two elements of $F$ yields again an element of $F$.
  • Addition is commutative, meaning that for any elements $a$ and $b$ of $F$, we have $a + b = b + a$.
  • Addition is associative, meaning that for any elements $a$, $b$, and $c$ of $F$, we have $a+(b+c) = (a+b)+c$. Thus we can write $a+b+c$ without any ambiguity.
  • Multiplication is also associative and commutative, meaning that for any elements $a$, $b$, and $c$ of $F$, we have $ab = ba$ and $a(bc) = (ab)c$ (and as above we can write $abc$ without any ambiguity).
  • Addition has an identity element, noted $0$, meaning that for any element $a$ of $F$, we have $0 + a = a + 0 = a$. $0$ is also often called the additive identity.
  • Every element of $F$ has an additive inverse, meaning that for any element $a$ of $F$, there exists an element $b$ of $F$ such that $a+b = b+a = 0$, the additive identity. The additive inverse of $a$ is noted $-a$.
  • Multiplication also has an identity element, noted $1$ and distinct from $0$, meaning that for any element $a$ of $F$, we have $1a = a1 = a$. $1$ is often called the multiplicative identity.
  • Every element of $F$ distinct from $0$ has a multiplicative inverse, meaning that for any non-zero element $a$ of $F$, there exists an element $b$ of $F$ such that $ab = ba = 1$, the multiplicative identity. The multiplicative inverse of $a$ is noted $a^{-1}$.
  • Finally, the two operations are related by the distributivity law, which states that for any elements $a$, $b$, and $c$ of $F$, we have $a(b+c) = ab+ac$.

Several important facts can be proven from the above properties:

  • For any element $a$ of $F$, we have $0a = 0$.
  • The cancellation law holds for addition, meaning that for any elements $a$, $b$, and $c$ of $F$, $a+b = a+c$ implies that $b = c$.
  • The cancellation law holds for multiplication if the cancelled term is not $0$, meaning that for any elements $a$, $b$, and $c$ of $F$, with $a \ne 0$, $ab = ac$ implies that $b = c$.

Also, the existence of additive and multiplicative inverses allow us to define the operations of substraction and division in the familiar way: $a – b = a + (-b)$ and $a/b = a(b^{-1})$ if $b \ne 0$.

0.2 Subfields and field extensions

If, given a field $E$, there exists a subset $F$ of $E$ such that the elements of $F$ form a field under the same operations as those in $E$, $F$ is said to be a subfield of $E$. A subfield which is a proper subset of $E$ (that is, whose elements are not all the elements of $E$) is called a proper subfield.

Given a field $E$ and a (not necessarily proper) subfield $F$ of $E$, we say that $E$ is an extension of $F$. More concisely, we say that $E/F$ is a field extension. Be careful to not interpret $E/F$ as a quotient or division of some sort.

Examples:

  • It is known that $\mathbf{R}$ is a subset of $\mathbf{C}$ (that is, all real numbers can also be viewed as complex numbers). Since $\mathbf{R}$ and $\mathbf{C}$ are both fields for the same operations of addition and multiplication, $\mathbf{R}$ is a subfield of $\mathbf{C}$, and $\mathbf{C}/\mathbf{R}$ is a field extension.
  • Also, $\mathbf{R}$ is a proper subset of $\mathbf{C}$, since the elements of $\mathbf{R}$ are not all of the elements of $\mathbf{C}$ (for example $i$ is an element of $\mathbf{C}$ but not of $\mathbf{R}$). Thus $\mathbf{R}$ is a proper subfield of $\mathbf{C}$.
  • $\mathbf{R}$ is a subset of $\mathbf{R}$, and so it is a subfield. Since it is not a proper subset, it is not a proper subfield. In general, any field $F$ is a subfield of itself, but not a proper subfield. $F$ is also the only subfield of $F$ that is not a proper subfield.
  • $\mathbf{C}$ is not a subfield of $\mathbf{R}$, since it is not a subset of $\mathbf{R}$.

0.3 Polynomials

The reader is probably already familiar with polynomials in real coefficients. Polynomials with coefficients in some field $F$ are the main tool in the study of the field extensions of $F$, and thus it is important to be comfortable with manipulating them.

For some field $F$, a polynomial in one indeterminate and with coefficients in $F$ is an expression of the form
\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \]
for some non-negative integer $n$, where all the coefficients $a_i$ are elements of $F$, and $a_n \ne 0$. The integer $n$ is called the degree of the polynomial. The set of all polynomials in one indeterminate and with coefficients in $F$ is noted $F[x]$. Polynomials can be added and multiplied in the usual fashion. Also, if $a_n = 1$, the polynomial is said to be monic.

Examples:

  • The polynomial $f(x) = x^2 + x – 1$ is a polynomial with real coefficients, but can also be viewed as a polynomial with complex coefficients.
  • The polynomial $g(x) = ix^2 + (2-i)x + \sqrt{2}$ is a polynomial in complex coefficients only.
  • For a real number $a$, we have the constant polynomial $h(x) = a$, which can be viewed as a polynomial in real or complex coefficients.
  • For a non-real complex number $z$, the constant polynomial $k(x) = z$ is a polynomial in complex coefficients only.

If $f(x)$ is a polynomial with coefficients in $F$ and $\alpha$ is an element of some extension $E$ of $F$, we can evaluate at $\alpha$ the polynomial $f(x)$ in the familiar manner: by replacing all occurrences of $x$ with $\alpha$, and performing all the necessary computations. The result is noted $f(\alpha)$, and is an element of $E$. When evaluating a polynomial $f(x)$ at some $\alpha \in E$, we also say we evaluate $f(x)$ in $E$.

Examples:

  • We can evaluate the above polynomial $f(x) = x^2+x-1$ at any element $\alpha$ of any extension of $\mathbf{R}$. For example we can evaluate it at $3 \in \mathbf{R}$, which yields $f(3) = 11$, again an element of $\mathbf{R}$.
  • We can also evaluate it at $1+i \in \mathbf{C}$, since $\mathbf{C}$ is an extension of $\mathbf{R}$. We have $f(1+i) = 3i$, an element of $\mathbf{C}$.
  • We can evaluate the above polynomial $g(x) = ix^2 + (2-i)x + \sqrt{2}$ in $\mathbf{C}$. We cannot evaluate it in $\mathbf{R}$, since multiplication by $i$ is an operation that takes place in $\mathbf{C}$, not in $\mathbf{R}$.

Next: Roots of a polynomial

Questions:

  • All the fields we have discussed so far ($\mathbf{R}$, $\mathbf{C}$, and we also mentioned $\mathbf{Q}$ in the introduction) have an infinite number of elements. Fields with only a finite number of elements also exist. Let $p$ be a prime number. Consider the set of $p$ elements
    \[ S_p = \{0, 1, 2, \dots , p-1\} \]
    and define the operations of addition and multplication as follows: first perform the usual addition or multiplication, and then take the remainder of the result when divided by $p$. It can be shown that $S_p$ with those two operations is a field, usually denoted $\mathbf{F}_p$. Verify that $\mathbf{F}_3$ is a field by checking that it satisfies all the properties of a field.
  • Recall that $\mathbf{R}[x]$ is the set of all polynomials with coefficients in $\mathbf{R}$. Since polynomials can be added and multiplied, it is natural to wonder whether $\mathbf{R}[x]$ is itself a field. The answer is no. Which property of a field does it fail to satisfy? How about $\mathbf{C}[x]$?

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