Fields

In abstract algebra, a field is a fundamental algebraic structure. It is a set on which addition, subtraction, multiplication, and division (by non-zero elements) are defined and behave as they do on the rational, real, and complex numbers. Fields are the building blocks for vector spaces and are crucial in many areas of mathematics, including number theory, algebraic geometry, and coding theory.

Fields of Fractions of Integral Domains

An integral domain is a commutative ring with unity (a multiplicative identity) and no zero divisors. This means that if $a$ and $b$ are elements of the integral domain, and $ab = 0$, then either $a = 0$ or $b = 0$. Examples of integral domains include the integers ($\mathbb{Z}$) and polynomials over a field.

The field of fractions (or quotient field) of an integral domain $D$ is the smallest field containing $D$. It is constructed by forming equivalence classes of fractions $\frac{a}{b}$, where $a$ and $b$ are elements of $D$ and $b \neq 0$. Two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are considered equivalent if $ad = bc$.

Construction of the Field of Fractions:

1. Define a set of ordered pairs $(a, b)$ where $a \in D$ and $b \in D \setminus \{0\}$.
2. Define an equivalence relation $\sim$ on this set: $(a, b) \sim (c, d)$ if and only if $ad = bc$.
3. The equivalence classes of this relation form the field of fractions, denoted by $F$. We represent the equivalence class of $(a, b)$ as $\frac{a}{b}$.
4. Addition is defined as: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$.
5. Multiplication is defined as: $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$.

The integral domain $D$ can be embedded into its field of fractions $F$ by identifying each element $a \in D$ with the equivalence class $\frac{a}{1}$.

Characteristics of a Field

The characteristic of a field $F$, denoted as $\text{char}(F)$, is the smallest positive integer $n$ such that $n \cdot 1 = 0$, where $1$ is the multiplicative identity of the field. If no such positive integer exists, the characteristic is said to be 0.

• If $n \cdot 1 = 0$ for some positive integer $n$, then $n$ must divide the characteristic. The characteristic is the smallest such positive integer.
• If $\text{char}(F) = n > 0$, then for any elements $a, b \in F$, we have $n \cdot (ab) = (n \cdot a)b = a(n \cdot b) = 0$.
• If $\text{char}(F) = n > 0$, then $n$ is also the characteristic of the additive subgroup generated by $1$.
• If $\text{char}(F) = 0$, then $n \cdot 1 \neq 0$ for all positive integers $n$.

An important property is that if a field $F$ has characteristic $p$ where $p$ is a prime number, then $F$ contains a subfield isomorphic to the field with $p$ elements, denoted by $\mathbb{F}_p$ or $\mathbb{Z}_p$. If the characteristic is 0, then $F$ contains a subfield isomorphic to the field of rational numbers $\mathbb{Q}$.

Examples of Characteristics:

• The field of rational numbers ($\mathbb{Q}$), real numbers ($\mathbb{R}$), and complex numbers ($\mathbb{C}$) all have characteristic 0. This is because no positive integer multiple of $1$ equals $0$ in these fields.
• The field $\mathbb{F}_p = \mathbb{Z}_p$ (integers modulo a prime $p$) has characteristic $p$. For example, in $\mathbb{F}_5$, we have $5 \cdot 1 = 1+1+1+1+1 = 0 \pmod{5}$.
• The field of rational functions $\mathbb{Q}(x)$ has characteristic 0.

Algebraic Extensions

Let $F$ and $K$ be fields such that $F \subseteq K$. We say that $K$ is an extension field of $F$, and $F$ is a subfield of $K$. An element $\alpha \in K$ is said to be algebraic over $F$ if there exists a non-zero polynomial $p(x)$ with coefficients in $F$ such that $p(\alpha) = 0$.

If every element of $K$ is algebraic over $F$, then $K$ is called an algebraic extension of $F$. If there is at least one element in $K$ that is not algebraic over $F$, then $K$ is called a transcendental extension of $F$.

Minimal Polynomial:

If $\alpha$ is algebraic over $F$, there is a unique monic (leading coefficient is 1) irreducible polynomial $m(x) \in F[x]$ of the smallest degree such that $m(\alpha) = 0$. This polynomial is called the minimal polynomial of $\alpha$ over $F$, denoted by $m_\alpha(x)$.

The set of all polynomials in $F[x]$ that have $\alpha$ as a root forms an ideal in $F[x]$. Since $F[x]$ is a principal ideal domain (PID), this ideal is generated by the minimal polynomial $m_\alpha(x)$.

Degree of an Extension:

If $\alpha$ is algebraic over $F$, the field extension $F(\alpha)$ (the smallest field containing $F$ and $\alpha$) is isomorphic to the quotient ring $F[x] / \langle m_\alpha(x) \rangle$. The degree of the extension $F(\alpha)$ over $F$, denoted by $[F(\alpha) : F]$, is equal to the degree of the minimal polynomial $m_\alpha(x)$.

If $F \subseteq K \subseteq L$ are fields, then $[L : F] = [L : K] [K : F]$. This property is multiplicative.

Example:

Consider the field of real numbers $\mathbb{R}$ and the complex number $i$. The polynomial $x^2 + 1 \in \mathbb{R}[x]$ has $i$ as a root, since $i^2 + 1 = -1 + 1 = 0$. The polynomial $x^2 + 1$ is irreducible over $\mathbb{R}$ and is monic. Therefore, $x^2 + 1$ is the minimal polynomial of $i$ over $\mathbb{R}$. The extension field $\mathbb{R}(i)$ is the field of complex numbers $\mathbb{C}$. The degree of the extension $[\mathbb{C} : \mathbb{R}]$ is 2, which is the degree of the minimal polynomial $x^2 + 1$.

The field $\mathbb{Q}(\sqrt{2})$ is an algebraic extension of $\mathbb{Q}$. The minimal polynomial of $\sqrt{2}$ over $\mathbb{Q}$ is $x^2 - 2$. Thus, $[\mathbb{Q}(\sqrt{2}) : \mathbb{Q}] = 2$.

Splitting Fields

A splitting field of a polynomial $p(x)$ over a field $F$ is the smallest field extension $K$ of $F$ in which $p(x)$ splits completely into linear factors. That is, $p(x)$ can be written as $p(x) = c(x - \alpha_1)(x - \alpha_2)...(x - \alpha_n)$, where $c \in F$ is the leading coefficient and $\alpha_1, \alpha_2, ..., \alpha_n$ are all elements of $K$.

Every polynomial over a field $F$ has a splitting field, and this splitting field is unique up to isomorphism over $F$. The splitting field is obtained by adjoining all the roots of the polynomial to the base field $F$.

Construction and Properties:

1. Let $p(x) \in F[x]$. Consider the roots of $p(x)$ in some extension field.
2. The splitting field $K$ is formed by $F$ and all the roots of $p(x)$. Formally, $K = F(\alpha_1, \alpha_2, ..., \alpha_n)$, where $\alpha_i$ are the roots of $p(x)$.
3. The splitting field $K$ is a finite extension of $F$. The degree of the extension $[K : F]$ depends on the polynomial.
4. If $p(x)$ is irreducible over $F$ and has degree $n$, then the splitting field $K$ is obtained by adjoining one root $\alpha$ to $F$, so $K = F(\alpha)$. In this case, $[K : F] = n$. However, this is only true if all roots of $p(x)$ are in $F(\alpha)$.

Examples:

• The splitting field of $p(x) = x^2 - 2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt{2})$. The roots are $\sqrt{2}$ and $-\sqrt{2}$. Adjoining $\sqrt{2}$ to $\mathbb{Q}$ gives $\mathbb{Q}(\sqrt{2})$, which also contains $-\sqrt{2}$.
• The splitting field of $p(x) = x^2 + 1$ over $\mathbb{R}$ is $\mathbb{C}$. The roots are $i$ and $-i$. Adjoining $i$ to $\mathbb{R}$ gives $\mathbb{R}(i) = \mathbb{C}$, which also contains $-i$.
• The splitting field of $p(x) = x^3 - 2$ over $\mathbb{Q}$. The roots are $\sqrt[3]{2}$, $\sqrt[3]{2}\omega$, and $\sqrt[3]{2}\omega^2$, where $\omega = e^{2\pi i / 3}$ is a primitive cube root of unity. The splitting field is $\mathbb{Q}(\sqrt[3]{2}, \omega)$. This is a degree 6 extension over $\mathbb{Q}$.

Simple Extensions

A field extension $K$ of $F$ is called a simple extension if $K$ can be obtained by adjoining a single element $\alpha$ to $F$. That is, $K = F(\alpha)$ for some element $\alpha$ in $K$.

If $\alpha$ is algebraic over $F$, then $F(\alpha)$ is a simple algebraic extension. Its degree is $[F(\alpha) : F] = \deg(m_\alpha(x))$, where $m_\alpha(x)$ is the minimal polynomial of $\alpha$ over $F$.

If $\alpha$ is transcendental over $F$, then $F(\alpha)$ is a simple transcendental extension. The field $F(\alpha)$ is isomorphic to the field of rational functions $F(x)$. The degree of a transcendental extension is infinite.

Examples:

• $\mathbb{Q}(\sqrt{2})$ is a simple algebraic extension of $\mathbb{Q}$, generated by $\sqrt{2}$.
• $\mathbb{R}(i)$ is a simple algebraic extension of $\mathbb{R}$, generated by $i$.
• $\mathbb{Q}(x)$, the field of rational functions in the variable $x$ with rational coefficients, is a simple transcendental extension of $\mathbb{Q}$, generated by $x$.
• $\mathbb{Q}(\sqrt{2}, \sqrt{3})$ is an algebraic extension of $\mathbb{Q}$, but it is not a simple extension. While it can be shown to be equal to $\mathbb{Q}(\sqrt{2} + \sqrt{3})$, demonstrating this requires more advanced techniques.

Primitive Element Theorem:

A significant result related to simple algebraic extensions is the Primitive Element Theorem. It states that if $K$ is a finite algebraic extension of a field $F$, and $F$ is a perfect field (e.g., fields of characteristic 0 or fields of characteristic $p$ where every element has a $p$-th root), then there exists an element $\alpha \in K$ such that $K = F(\alpha)$. This means that every finite algebraic extension over a perfect field is a simple extension.