Galois Theory: Field Theory Fundamentals

The Fundamental Theorem of Algebra

• Every non-constant, single-variable polynomial with complex coefficients has at least one complex root.
• If a polynomial $p$ has degree $n$ over $\mathbb{C}$, then $p$ has exactly $n$ roots, counted with multiplicity.

Galois Theory

• Initiated by Évariste Galois (1811-1832).
• Studies solutions of polynomial equations, $p(x) = 0$.
• Built upon the work of Lagrange, Abel, and Ruffini.

Field Theory for Galois Theory

• A field $F$ is a set with + and · operations satisfying standard axioms.
• Examples of fields include $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$ where $p$ is prime.
• $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$ (when $n$ is not prime) are not fields.
• If $E$ and $F$ are fields and $E \subseteq F$, then $F$ is an extension field of $E$.
• $\mathbb{C}$ is an extension field of $\mathbb{R}$.
• A field $F$ contains a subfield isomorphic to either $\mathbb{Q}$ or $\mathbb{F}_p$ for some prime $p$, called the prime subfield of $F$.
  • If the prime subfield is $\mathbb{Q}$, $F$ has characteristic zero.
  • If the prime subfield is $\mathbb{F}_p$, $F$ has characteristic $p$.
• Given a subfield $F$ of $E$ and $\alpha \in E$, $\alpha$ is algebraic over $F$ if it is a root of a non-zero polynomial $f(x) \in F[x]$.
  • $\sqrt{2}$ is algebraic over $\mathbb{Q}$ because it is a root of $x^2 - 2$.
  • $i$ is algebraic over $\mathbb{R}$ because it is a root of $x^2 + 1$.
  • $\pi$ is not algebraic over $\mathbb{Q}$.
• If $\alpha$ is algebraic over $F$, there is a unique monic polynomial $m(x) \in F[x]$ of minimal degree such that $m(\alpha) = 0$; $m(x)$ is the minimal polynomial of $\alpha$ over $F$.
• A field extension $E$ of $F$ is an algebraic extension if every element of $E$ is algebraic over $F$.
• A field extension $E$ of $F$ is a finite extension if $E$ is a finite-dimensional vector space over $F$, denoted as $[E:F]$.
  • If $E$ is a finite extension of $F$, then $E$ is an algebraic extension of $F$, but the converse is not always true.
• For $F = \mathbb{Q}$, and $E = \mathbb{Q}(\sqrt{2}) = {a + b\sqrt{2} : a, b \in \mathbb{Q}}$, $E$ is a field and an extension field of $F$.
  • $[E:F] = 2$ since ${1, \sqrt{2}}$ is a basis for $E$ over $F$.
• For a field $F$ and $f(x) \in F[x]$, an extension field $E$ of $F$ is a splitting field of $f(x)$ over $F$ if:
  • $f(x)$ factors completely into linear factors in $E[x]$, and
  • $f(x)$ does not factor completely into linear factors in any proper subfield of $E$ containing $F$.
• For $F = \mathbb{Q}$ and $f(x) = x^2 - 2$, $E = \mathbb{Q}(\sqrt{2})$ is a splitting field of $f(x)$ over $F$.
• For a field $F$ and $f(x) \in F[x]$, a splitting field of $f(x)$ over $F$ exists.

Group Theory for Galois Theory

• A group $G$ is a set with an operation * satisfying closure, associativity, identity, and inverse axioms.
• A subgroup of $G$ is a subset of $G$ which forms a group under the same operation as $G$, denoted as $H \le G$.
• For $H \le G$, a left coset of $H$ in $G$ is a set $gH = {gh : h \in H}$ for some $g \in G$.
• If $G$ is a finite group and $H \le G$, then $|H|$ divides $|G|$ (Lagrange's Theorem).
• $H$ is a normal subgroup of $G$ if $gH = Hg$ for all $g \in G$, denoted as $H \trianglelefteq G$.
• If $H \trianglelefteq G$, the set of cosets of $H$ in $G$ forms the quotient group $G/H$ under the operation $(aH)(bH) = (ab)H$.

Up Next:

• Reviewing field and group theory will be completed.
• Defining Galois groups.
• Fundamental Theorem of Galois Theory will be stated.
• Abel-Ruffini Theorem will be proven. The theorem states that there is no general algebraic formula for polynomial roots of degree 5 or higher.