Chapter 13: Field Theory

13.1 Basic Theory of Field Extensions

Definition: The characteristic of a field $F$, denoted $\mathrm{ch}(F)$, is the smallest $p \in \mathbb{Z}^{+}$ such that $p\cdot1_{F}=0$ if such a $p$ exists. Otherwise, $\mathrm{ch}(F)=0$.
Proposition 1: $\mathrm{ch}(F)$ is either $0$ or a prime $p$. If $\mathrm{ch}(F)=p$ then for any $\alpha \in F$, $\alpha p=0$.
Fact: This notion of a characteristic makes sense for an integral domain too, whose characteristic will be the same as for its fields of fractions.
Definition: The prime subfield of a field $F$ is the subfield of $F$ generated by the multiplicative identity $1_{F}$ of $F$. It is isomorphic to $\mathbb{Q}$ if $\mathrm{ch}(F)=0$ or $\mathbb{F}_{p}$ if $\mathrm{ch}(F)=p$.
Definition: If $K$ is a field containing the subfield $F$, then $K$ is said to be an extension field or simply an extension of $F$, denoted $K/F$ or by the diagram

In particular, every field is an extension of its prime subfield. The field $F$ is sometimes called the base field of the extension.
Fact: If $K/F$ is any extension of fields, then the multiplication defined in $K$ makes $K$ a vector space over $F$. In particular every field $F$ can be considered a vector space over its prime field.
Definition: The degree (or relative degree or index) of a field extension $K/F$, $[K:F]$, is the dimension of $K$ as a vector space over $F$ (i.e., $[K:F]=\mathrm{dim}_{F}K$). The extension is said to be finite if $[K:F]$ is finite and is otherwise infinite.
Proposition 2: Let $\varphi:F\to F'$ be a field homomorphism. Then $\varphi$ is either identically $0$ or is injective, i.e. $\varphi(F)\cong \{ 0 \}$ or $\varphi(F)\cong F$.
Theorem 3: Let $F$ be a field and let $p(x)\in F[x]$ be an irreducible polynomial. Then there exists a field $K$ containing an isomorphic copy of $F$ in which $p(x)$ has a root. Identifying $F$ with this isomorphic copy shows that there exists an extension of $F$ in which $p(x)$ has a root.
Theorem 4: Let $p(x)\in F[x]$ be an irreducible polynomial of degree $n$ over the field $F$ and let $K$ be the field $F[x]/(p(x))$. Let $\theta=x\; (\text{mod} \; (p(x)) )\in K$. Then the elements $1,\theta,\dots,\theta^{n-1}$ are a basis for $K$ as a vector space over $F$, so $[K:F]=n$. Hence $K=\{ a_{0}+a_{1}\theta+\dots+a_{n-1}\theta^{n-1} \mid a_{0},a_{1},\dots,a_{n-1}\in F \}$, i.e. all polynomials of degree $<n$ in $\theta$.
Corollary 5: Let $K$ be as in Theorem 4 and let $a(\theta),b(\theta)\in K$ be two polynomials of degree $<n$ in $\theta$. Then addition in $K$ is analogous to usual polynomial addition and multiplication in $K$ is defined by $a(\theta)b(\theta)=r(x)$, where $r(x)=a(x)b(x)\; (\text{mod} \; p(x) )$.
Definition: Let $K$ be an extension of the field $F$ and let $\alpha,\beta,\dots \in K$. Then the smallest subfield of $K$ containing both $F$ and the elements $\alpha,\beta,\dots$ is called the field generated by $\alpha,\beta,\dots$ over F and denoted $F(\alpha,\beta,\dots)$.
Definition: If the field $K$ is generated by a single element $\alpha$ over $F$, $K=F(\alpha)$, then $K$ is said to be a simple extension of $F$ and the element $\alpha$ is called a primitive element for the extension.
Fact: Every finite extension of a field of characteristic $0$ is a simple extension.
Theorem 6: Let $F$ be a field and let $p(x)\in F[x]$ be irreducible. Suppose $K$ extends $F$ and contains a root $\alpha$ of $p(x)$. Then $F(\alpha)\cong F[x]/(p(x))$. In particular, any field over $F$ in which $p(x)$ contains a root must contain a subfield isomorphic to the extension of $F$ constructed in Theorem 3 and that this field is, up to isomorphism, the smallest extension of $F$ containing such a root.
Corollary 7: Suppose in Theorem 6 that $\mathrm{deg\;}p(x)=n$. Then $F(\alpha)=\{ a_{0}+a_{1}\alpha+\dots+a_{n-1}\alpha^{n-1}\mid a_{0},a_{1},\dots,a_{n-1}\in F \}\subseteq K$.
Fact: The roots of an irreducible polynomial $p(x)\in F[x]$ are algebraically indistinguishable in the sense that the fields obtained by adjoining any root of an irreducible polynomial are isomorphic.
Theorem 8: Let $\varphi:F\ \tilde{\longrightarrow}\ F'$ be a field isomorphism. Let $p(x)\in F[x]$ be be irreducible and let $p'(x)\in F'[x]$ be the polynomial obtained by applying $\varphi$ to the coefficients of $p(x)$. Let $\alpha$ be a root of $p(x)$ in some extension of $F$ and let $\beta$ be a root of $p'(x)$ in some extension of $F'$. Then there exists an isomorphism $\sigma:F(\alpha)\ \tilde{\longrightarrow}\ F'(\beta)$ such that $\alpha\longmapsto\beta$ and such that $\sigma$ extends $\varphi$, i.e. $\sigma$ restricted to $F$ is the isomorphism $\varphi$. Pictorially, it is

13.2 Algebraic Extensions

Let $F$ be a field and $K$ be an extension of $F$.
Definition: The element $\alpha \in K$ is algebraic over $F$ if $f(\alpha)=0$ for some nonzero $f(x)\in F[x]$. If $\alpha$ is not algebraic over $F$, then $\alpha$ is transcendental over $F$. $K/F$ is algebraic if every element of $K$ is algebraic over $F$.
Fact: If $\alpha$ is algebraic over $F$ then it is trivially algebraic over any extension field $L$ of $F$.
Proposition 9: Let $\alpha$ be algebraic over $F$. Then there exists a unique monic irreducible polynomial $m_{\alpha,F}(x)\in F[x]$ for which $\alpha$ is a root. $f(x)\in F[x]$ has $f(\alpha)=0\iff m_{\alpha,F}(x) \mid f(x)$ in $F[x]$.
Corollary 10: If $L/F$ is an extension of fields and $\alpha$ is algebraic over both $F$ and $L$, then $m_{\alpha,L}(x) \mid m_{\alpha,F}(x)$ in $L[x]$.
Definition: The polynomial $m_{\alpha,F}(x)$ is called the minimal polynomial for $\alpha$ over $F$. The degree of $m_{\alpha}(x)$ is also called the degree of $\alpha$.
Proposition 11: Let $\alpha$ be algebraic over $F$. Then $F(\alpha)\cong F[x]/(m_{\alpha}(x))$, and thus $[F(\alpha):F]=\mathrm{deg\;}m_{\alpha}(x)=\mathrm{deg\;}\alpha$, i.e. the degree of $\alpha$ over $F$ is the degree of the extension it generates over $F$.
Proposition 12: The element $\alpha$ is algebraic over $F$ $\iff$ the simple extension $F(\alpha)/F$ is finite. More precisely, if $\alpha$ is an element of an extension of degree $n$ over $F$ then $\alpha$ is a root of a polynomial of degree at most $n$ over $F$, and if $\alpha$ satisfies a polynomial of degree $n$ over $F$ then the degree of $F(\alpha)$ over $F$ is at most $n$.
Corollary 13: If the extension $K/F$ is finite, then it is algebraic.
Fact: Any extension $K$ of $F$ such that $[K:F]=2$ is of the form $F(\sqrt{ D })$, where $D\in F$ such that $D$ is not a square in $F$, and conversely every extension of the form $F(\sqrt{ D })$ has degree $2$. These are known as the quadratic extensions of $F$.
Theorem 14: Let $F\subseteq K\subseteq L$ be fields. Then $[L:F]=[L:K][K:F]$, i.e. extension degrees are multiplicative, and if one side of the equation is infinite, the other side is also infinite. Pictorially,

Corollary 15: Suppose $L/F$ is a finite extension and let $F\subseteq K\subseteq L$. Then $[K:F] \mid [L:F]$.
Definition: An extension $K/F$ is finitely generated if $\exists\alpha_{1},\dots,\alpha_{k}\in K$ such that $K=F(\alpha_{1},\dots,\alpha_{k})$.
Lemma 16: $F(\alpha,\beta)=(F(\alpha))(\beta)$, i.e. the field generated over $F$ by $\alpha$ and $\beta$ is the field generated by $\beta$ over the field $F(\alpha)$ generated by $\alpha$.
Fact: The field $F(\alpha)$ for algebraic $\alpha$ is typically obtained by adjoining $\alpha$ to $F$ and then closing the resulting set with respect to addition and multiplication. The process terminates when a power of $\alpha$ is a linear combination of lower powers of $\alpha$, i.e. finding the minimal polynomial for $\alpha$. If $\alpha$ is non-algebraic, it is necessary to also close with respect to additive and multiplicative inverse.
Theorem 17: The extension $K/F$ is finite $\iff$ $K$ is finitely finitely generated by algebraic elements over of $F$. More precisely, let $\alpha_{1},\dots,\alpha_{k}\in F$ be algebraic with degrees $n_{1},\dots,n_{k}$. Then $\mathrm{deg\;}F(\alpha_{1},\dots,\alpha_{k})\leq n_{1}\dots n_{k}$.
Corollary 18: Suppose $\alpha,\beta$ are algebraic over $F$. Then $\alpha\pm\beta,\alpha\beta,\alpha/\beta\ (\text{for }\beta\neq0)$ are all algebraic.
Corollary 19: Let $L/F$ be an arbitrary extension. Then the set of elements of $L$ that are algebraic over $F$ form a subfield $K$ of $L$.
Theorem 20: If $K$ is algebraic over $F$ and $L$ is algebraic over $K$, then $L$ is algebraic over $F$.
Definition: Let $K_{1},K_{2}\subseteq K$. Then the composite field of $K_{1}$ and $K_{2}$, denoted $K_{1}K_{2}$, is the smallest subfield of $K$ with $K_{1},K_{2}\subseteq K_{1}K_{2}$. This extends to any number of subfields.
Proposition 21: Let $K_{1},K_{2}$ be two finite extensions of a field $F\subseteq K$. Then $[K_{1}K_{2}:F]\leq[K_{1}:F][K_{2}:F]$. Equality is true $\iff$ an $F$-basis for one of the fields remains linearly independent over the other field. If $\{ \alpha_{1},\dots,\alpha_{n} \}$ and $\{ \beta_{1},\dots,\beta_{m} \}$ are the bases for $K_{1}$ and $K_{2}$, respectively, then the elements $\alpha_{i}\beta_{j}$ span $K_{1}K_{2}$ over $F$. Pictorially, Corollary 22: Suppose that $[K_{1}:F]=n$, $[K_{2}:F]=m$ in Proposition 21, where $(m,n)=1$. Then $K_{1}K_{2}:F=[K_{1}:F][K_{2}:F]=mn$.

13.3 Classical Straightedge and Compass Constructions

Proposition 23: If the element $\alpha \in \mathbb{R}$ is obtained from a field $F\subset \mathbb{R}$ by a series of compass and straightedge constructions then $[F(\alpha):F]=2^{k}$ for some integer $k\geq 0$.
Theorem 24: None of the classical Greek problems: (I) Doubling the Cube, (II) Trisecting and Angle, and (III) Squaring the Circle, is possible.

13.4 Splitting Fields and Algebraic Closures

Let $F$ be a field.
Definition: The extension field $K$ of $F$ is called a splitting field for the polynomial $f(x)\in F[x]$ if $f(x)$ factors (splits) completely into linear factors in $K[x]$ and $f(x)$ does not factor completely over any proper subfield of $K$ containing $F$.
Theorem 25: If $f(x)\in F[x]$ then there exists an extension $K$ of $F$ which is a splitting field for $f(x)$.
Fact: If $f(x)\in F[x]$ such that $\mathrm{deg\;}f(x)=1$, then the splitting field of $F$ for $f(x)$ is simply $F$. If $\mathrm{deg\;}f(x)=2$ and $f(x)$ is irreducible, then the algebraic extension $F(\alpha)$ over $F$ for some root $\alpha$ of $f(x)$ contains all roots of $f(x)$. If $\mathrm{deg\;}f(x)\geq 3$, then the splitting field may be a chain of algebraic extensions.
Definition: If $K$ is an algebraic extension of $F$ which is the splitting field over $F$ for a collection of polynomials $f(x)\in F[x]$ then $K$ is called a normal extension of $F$.
Proposition 26: A splitting field of a polynomial of degree $n$ over $F$ is of degree at most $n!$ over $F$.
Definition: A generator of the cyclic group of all $n$th roots of unity is called a primitive $n$th root of unity $\zeta_{n}$.
Definition: The field $\mathbb{Q}(\zeta_{n})$ is called the cyclotomic field of $n$th roots of unity.
Fact: The polynomial $\frac{x^{p}-1}{x-1}$ is the minimal polynomial for $\zeta_{p}$ and has degree $\varphi(p)=p-1$. In general, $[\mathbb{Q}(\zeta_{n}):\mathbb{Q}]=\varphi(n)$.
Theorem 27: Let $\varphi:F \tilde{\to} F'$ be a field isomorphism. Let $f(x)\in F[x]$ be a polynomial and let $f'(x)\in F'[x]$ be the polynomial obtained by applying $\varphi$ to the coefficients of $f(x)$. Let $E$ be a splitting field for $f(x)$ over $F$ and let $E'$ be a splitting field for $f'(x)$ over $F'$. Then the isomorphism $\varphi$ extends to an isomorphism $\sigma:E \tilde{\to}E'$, i.e. $\sigma$ restricted to $F$ is $\varphi$. Pictorially, Corollary 28: (Uniqueness of Splitting Fields) Any two splitting fields for a polynomial $f(x)\in F[x]$ over a field $F$ are isomorphic.
Definition: The field $\overline{F}$ is called an algebraic closure of $F$ is $\overline{F}$ is algebraic over $F$ and if every polynomial $f(x)\in F[x]$ splits completely over $\overline{F}$ (that is, $\overline{F}$ contains all the elements algebraic over $F$).
Definition: A field $K$ is algebraically closed if every polynomial with coefficients in $K$ has a root in $K$.
Proposition 29: Let $\overline{F}$ be an algebraic closure of $F$. Then $\overline{F}$ is algebraically closed.
Proposition 30: For any field $F$ there exists an algebraically closed field $K$ containing $F$.
Proposition 31: Let $K$ be an algebraically closed field and let $F$ be a subfield of $K$. Then the collection of elements $\overline{F}$ of $K$ that are algebraic over $F$ is an algebraic closure of $F$. An algebraic closure of $F$ is unique up to isomorphism.
Theorem: (Fundamental Theorem of Algebra) The field $\mathbb{C}$ is algebraically closed.
Corollary 32: The field $\mathbb{C}$ contains an algebraic closure for any of its subfields. In particular, $\overline{\mathbb{Q}}$, the collection of complex numbers algebraic over $\mathbb{Q}$, is an algebraic closure of $\mathbb{Q}$.