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Basics of field theory

Things to remember from before.

We already know quite a bit about fields.

Characteristic

If $F$ is a field, then there is a ring homomorphism $\Z\to F$ sending $1\to 1$. If this map is injective, then:

  • we say $F$ has characteristic zero
  • $F$ contains a copy of the rational numbers
  • The field $\Q$ is the prime subfield of $F$.

Otherwise the kernel of this map must be a prime ideal $p\Z$ of $\Z$. In this case:

  • we say that $F$ has characteristic $p$
  • $F$ contains a copy of $\Z/p\Z$.
  • $\Z/p\Z$ is the prime subfield of $F$.

Maps

If $f:F\to E$ is a homomorphism of fields, it is automatically injective (or zero).

The only field maps $f:\Q\to \Q$ and $f:\Z/p\Z\to \Z/p\Z$ are the identity.

Extensions

If $F$ is a field, and $F\subset E$ where $E$ is another field, then we call $E$ an extension field of $F$.

$E$ is automatically a vector space over $F$. The degree of $E/F$, written $[E:F]$, is the dimension of $E$ as an $F$-vector space.

Polynomials, quotient rings, and fields

We have the division algorithm for polynomials. $F[x]$ is a PID. An ideal is prime iff it is generated by an irreducible polynomial.

Let $p(x)$ be an irreducible polynomial of degree $d$ over $F$. Then:

  • $K=F[x]/(p(x))$ is a field
  • It is of degree $d$ over $F$.
  • $p(x)$ has a root in $K$ (namely the residue class of $x$)
  • The elements $1,x,\ldots, x^{d-1}$ are a basis for $K/F$.

Adjoining roots of polynomials

If $F\subset K$ is a field extension, and $\alpha\in K$, then $F(\alpha)$ is the smallest subfield of $K$ containing $F$ and $\alpha$. Similarly for $F(\alpha_1,\alpha_2,\ldots,\alpha_n)$.

If $p(x)$ is irreducible over $F$, and has a root $\alpha$ in $K$, then $F(\alpha)$ is isomorphic to $F[x]/p(x)$ via the map $x\mapsto \alpha$.

Key Theorem

Let $K$ be a field extension of $F$ and let $p(x)$ be an irreducible polynomial over $F$. Suppose $K$ contains two roots $\alpha$ and $\beta$ of $p(x)$. Then $F(\alpha)$ and $F(\beta)$ are isomorphic via an isomorphism that is the identity on $F$.

More generally:

Theorem: (See Theorem 8, DF, page 519) Let $\phi:F\to F’$ be an isomorphism of fields. Let $p(x)$ be an irreducible polynomial in $F[x]$ and let $p’(x)$ be the polynomial in $F’[x]$ obtained by applying $\phi$ to the coefficients of $p(x)$. Let $K$ be an extension of $F$ containing a root $\alpha$ of $p(x)$, and let $K’$ be an extension of $F’$ containing a root $\beta$ of $p’(x)$. Then there is an isomorphism $\sigma:F(\alpha)\to F’(\beta)$ such that the restriction of $\sigma$ to $F$ is $\phi$.

Algebraic Extensions

Definition

Definition: Let $F\subset K$ be a field extension. An element $\alpha\in K$ is algebraic over $F$ if it is the root of a nonzero polynomial in $F[x]$. Elements that aren’t algebraic are called transcendental.

An extension $K/F$ is algebraic if every element of $K$ is algebraic over $F$.

Basics

  • If $\alpha$ is algebraic over $F$, there is unique monic polynomial $m_{\alpha,F}(x)$ of minimal degree with coefficients in $F$ such that $m_{\alpha}(\alpha)=0$. (This follows from the division algorithm). This polynomial is called the minimal polynomial of $\alpha$ over $F$. Its degree is the degree of $\alpha$.
  • If $F\subset L$, then the minimal polynomial $m_{\alpha,L}(x)\in L[x]$ of $\alpha$ over $L$ divides the minimal polynomial $m_{\alpha,F}(x)$. Again, this follows from the division algorithm for $L[x]$.
  • $F(\alpha)$ is isomorphic to $F[x]/m_{\alpha,F}(x)$; and the degree $[F(\alpha):F]$ is the degree of $\alpha$.

Examples

If $n>1$ and $p$ is a prime, then the polynomial $x^{n}-p$ is irreducible over $\Q$, so $\alpha=\sqrt[n]{p}$ has degree $n$ over $\Q$.

The polynomial $x^3-x-1$ is irreducible over $\Q$ and has one real root $\alpha$. So $\alpha$ has degree $3$ over $\Q$ but degree $1$ over $\R$.

Finite extensions are algebraic

Suppose $K/F$ is finite and let $\alpha$ be an element of $K$. Then there is an $n$ so that the set $1,\alpha,\alpha^2,\ldots, \alpha^{n}$ are linearly dependent over $F$; so $\alpha$ satisfies a polynomial with $F$ coefficients, and is therefore algebraic.

As a partial converse, if $F(\alpha)/F$ is finite if and only if $\alpha$ is algebraic. If $\alpha$ is algebraic of degree $d$ over $F$, $F(\alpha)=F[x]/(m_{\alpha}(x))$ which is finite dimensional (with basis $1,x,x^2,\ldots, x^{d-1}$.)

Field Degrees

Multiplicativity of degrees

Proposition: Suppose that $L/F$ and $K/L$ are extensions. Then $[K:F]=[K:L][L:F]$.

Proof: If $\alpha_1,\ldots, \alpha_n$ are a basis for $L/F$, and $\beta_1,\ldots, \beta_k$ are a basis for $K/L$, then the products $\alpha_{i}\beta_{j}$ are a basis for $K/F$.

Corollary: If $L/F$ is a subfield of $K/F$, then $[L:F]$ divides $[K:F]$.

Finitely generated extensions

A field $K/F$ is finitely generated if $K=F(\alpha_1,\ldots, \alpha_n)$ for a finite set of $\alpha_{i}$ in $K$.

Proposition: $F(\alpha,\beta)=F(\alpha)(\beta)$.

Proof: $F(\alpha,\beta)$ contains $F(\alpha)$ and also $\beta$. Therefore $F(\alpha)(\beta)\subset F(\alpha,\beta)$. On the other hand, since $\alpha$ and $\beta$ are in $F(\alpha)(\beta)$, we know that $F(\alpha,\beta)\subset F(\alpha)(\beta)$.

Finite is finitely generated

Proposition: A field $K/F$ is finite if and only if it is finitely generated. If it is generated by $\alpha_1,\ldots, \alpha_k$ then it is of degree at most $n_1 n_2\ldots n_k$ where $n_i$ is the degree of $\alpha_i$ over $F$.

Proof: If it’s finitely generated, then it’s a sequence of extensions $F(\alpha_1,\ldots, \alpha_{s-1})(\alpha_{s})$ each of degree at most $n_i$. So $K/F$ is finite. Conversely, if $K/F$ is finite (and of degree greater than 1), choose $\alpha_1\in K$ of degree greater than 1. Then $F(\alpha)\subset K$ and $[K:F(\alpha)]$ is smaller than $[K:F]$. Now choose $\alpha_2$ in $K$ but not $F(\alpha_1)$, and so on. This process must terminate.

Corollary: If $\alpha$ and $\beta$ are algebraic over $F$, so are $\alpha+\beta$, $\alpha\beta$, and (if $\beta\not=0$) $\alpha/\beta$.

Proof: All these elements lie in $F(\alpha,\beta)$ which is finite over $F$.

Corollary: If $K/F$ is a field extension, the subset of $K$ consisting of algebraic elements over $F$ is a field (called the algebraic closure of $F$ in $K$).

Towers of algebraic extensions are algebraic

Propositoin: If $L/K$ is algebraic and $K/F$ is algebraic so is $L/F$.

Proof: Choose $\alpha\in L$. Then $\alpha$ satisfies a polynomial $f(x)=x^{d}+a_{d-1}x^{d-1}+\cdots+a_{0}$ where the $a_i$ are in $K$. Therefore $\alpha$ is algebraic over $E=F(a_0,a_1,\ldots, a_{d-1})$. But $E/F$ is finitely generated hence finite. Therefore $[E(\alpha):F]=[E(\alpha):E][E:F]$ is finite. Thus every element of $L$ is algebraic over $F$.

Composites

If $K_1$ and $K_2$ are subfields of a field $K$, then $K_1 K_2$ is the smallest subfield of $K$ containing these two fields. Then $[K_1 K_2:F]$ is divisible by both $[K_1:F]$ and $[K_2:F]$ and in addition

\[[K_1 K_2:F]\le [K_1:F][K_2:F].\]

In particular, if $[K_1:F]$ and $[K_2:F]$ are relatively prime, then $[K_1 K_2:F]=[K_1:F][K_2:F]$.

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