Abstract Algebra, Judson

Created: October 28, 2024
Updated: November 09, 2024

A free-to-read online textbook about abstract algebra, written by Thomas W. Judson.

Table of Contents

Chapter 10: Normal Subgroups and Factor Groups

  • Factor Groups and Normal Subgroups
    • Definition: $H$ is a normal subgroup of $G$ if $gH = Hg$ for all $g \in G$.
    • Example: Every subgroup of an abelian group is normal.
    • Example: The subgroup ${(1), (12)}$ of $S _ 3$ is not normal since $(123)H \ne H(123)$
    • Theorem: The following are equivalent
      • $N$ is a normal subgroup of $G$
      • For all $g \in G$, $g N g^{-1} \subset N$
      • For all $g \in G$, $g N g^{-1} = N$
    • Theorem: If $N \trianglelefteq G$, then the cosets of $N$ in $G$ form a group $G/N$ of order $[G:N]$.
    • Example: $S _ 3 / A _ 3 \cong C _ 2$
    • Example: $\mathbb Z / n \mathbb Z$
    • Example: $D _ n / R _ n \cong C _ 2$
  • The Simplicity of the Alternating Group
    • F: Definition: Groups with no nontrivial normal subgroups are called simple groups.
    • Example: $\mathbb Z / p\mathbb Z$ is simple for $p$ prime, trivially simple since no nontrivial subgroups, let alone nontrivial normal subgroups
    • Lemma: $A _ n$ is generated by $3$-cycles for $n \ge 3$
    • F: Lemma: If $N \trianglelefteq A _ n$ for $n \ge 3$ and $N$ contains a three cycle, then $N = A _ n$.
    • F: Lemma: For $n \ge 5$, every nontrivial normal subgroup $N$ of $A _ n$ contains a 3-cycle
    • F: Theorem: The alternating group $A _ n$ is simple for $n \ge 5$

Chapter 13: The Structure of Groups

  • Finite Abelian Groups
    • Definition: The group generated by ${g _ i : i \in I}$ is the smallest group containing all elements, and the $g _ i$s are said to be the generators.
    • Definition: If $G = \langle g _ 1, \ldots, g _ n \rangle$ finite, then $G$ is finitely generated
    • Example: All finite groups are finitely generated, but e.g. $\mathbb Z \times \mathbb Z _ n$ is an infinite group but finitely generated
    • Example: $\mathbb Q$ is not finitely generated
    • Proposition: If $G = \langle g _ i : i \in I\rangle$, then every element can be written as a product of powers of group elements
    • Definition: A group $G$ is a $p$-group if the order of every element of $G$ is a power of $p$.
    • Theorem: Fundamental Theorem of Finite Abelian Groups: Every finite abelian group $G$ is isomorphic to a direct product of cyclic groups of prime-power order.
    • Lemma: If $G$ is a finite abelian group of order $n$, and $p$ is a prime such that $p \mid n$, then $G$ contains an element of order $p$
    • Lemma: A finite abelian group is a $p$-group iff its order is a power of $p$
    • Lemma: Let $G$ be a finite abelian group of order $n = \prod p _ i^{\alpha _ i}$ where $p _ i$ are prime and $\alpha _ i$ positive integers. Then $G$ is the internal direct product of subgroups $G _ 1, \ldots, G _ k$ where $G _ i$ is the subgroup of $G$ consisting of all elements of order $p _ i^r$ for some integer $r$.
    • Lemma: Let $G$ be a finite abelian $p$-group and suppose $g \in G$ has maximal order. Then $G$ is isomorphic to $\langle g \rangle \times H$ for some subgroup $H$ of $G$.
    • Theorem: Fundamental Theorem of Finitely Generated Abelian Groups: Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups of the form $\mathbb Z _ {p _ 1}^{\alpha _ 1} \times \mathbb Z _ {p _ 2}^{\alpha _ 2} \times \cdots \times \mathbb Z _ {p _ n}^{\alpha _ n} \times \mathbb Z \times \cdots \times \mathbb Z$
  • Solvable Groups
    • F: Definition: A subnormal series of a group $G$ is a finite sequence of subgroups $G = H _ n \supset H _ {n-1} \supset \cdots \supset H _ 1 \supset H _ 0 = {e}$ where $H _ i$ is a normal subgroup of $H _ {i+1}$.
    • F: Definition: The series is normal if each subgroup $H _ i$ is normal in $G$
    • Definition: The length of a normal series is the number of proper inclusions
    • F: Example: Any series of subgroups of an abelian group is a normal series, e.g:
      • $\mathbb Z \supset 9\mathbb Z \supset 45 \mathbb Z \supset 180 \mathbb Z \supset {0}$
      • $\mathbb Z _ {24} \supset \langle 2 \rangle \supset \langle 6 \rangle \supset \langle 12 \rangle \supset {0}$
    • Example: $D _ 4 \supset {(1), (12)(34), (13)(24), (14)(23)} \supset {(1),(12)(34)} \supset {e}$ is a subnormal series that is not normal
    • Definition: A subnormal series ${K _ j}$ is a refinement of a subnormal series ${H _ i}$ if ${H _ i} \subset {K _ j}$, i.e. each $H _ i$ appears in ${K _ j}$
    • Example: $\mathbb Z \supset 3 \mathbb Z \supset 9 \mathbb Z \supset \mathbb 45 \supset 90 \mathbb Z \supset 180 \mathbb Z \supset {0}$ is a refinement of $\mathbb Z \supset 9 \mathbb Z \supset 45 \mathbb Z \supset 180\mathbb Z \supset {0}$
    • Remark: It’s useful to study the factor groups $H _ {i+1} / H _ i$
    • F: Definition: Two subnormal series ${H _ i}$ and ${K _ j}$ of a group $G$ are isomorphic if there is a one-to-one correspondence between ${H _ {i+1} / H _ i}$ and ${K _ {j+1} / K _ j}$.
    • Example:
      • $\mathbb Z _ {60} \supset \langle 3 \rangle \supset \langle 15 \rangle \supset {0}$
      • $\mathbb Z _ {60} \supset \langle 4 \rangle \supset \langle 20 \rangle \supset {0}$
      • These are isomorphic since:
      • $\mathbb Z _ {60} / \langle 3 \rangle \cong \langle 20 \rangle / \langle 0 \rangle \cong \mathbb Z _ 3$
      • $\langle 3 \rangle / \langle 15 \rangle \cong \langle 4 \rangle / \langle 20 \rangle \cong \mathbb Z _ 5$
      • $\langle 15 \rangle / {0} \cong \mathbb Z _ {60} / \langle 4 \rangle \cong \mathbb Z _ 4$
    • F: Definition: A subnormal series ${H _ i}$ of a group $G$ is a composition series if all the factor groups are simple, equivalently none of the factor groups contains a normal subgroup
    • F: Definition: A normal series ${H _ i}$ of $G$ is called a principal series if all the factor groups are simple
    • Example:
      • $\mathbb Z _ {60}$ has a composition series $\mathbb Z _ {60} \supset \langle 3 \rangle \supset \langle 15 \rangle \supset \langle 30 \rangle \supset {0}$
      • The factor groups are:
      • $\mathbb Z _ {60} / \langle 3 \rangle \cong \mathbb Z _ 3$
      • $\langle 3 \rangle / \langle 15 \rangle \cong \mathbb Z _ 5$
      • $\langle 15 \rangle / \langle 30 \rangle \cong \mathbb Z _ 2$
      • $\langle 30 \rangle / {0} \cong \mathbb Z _ 2$
      • Since $\mathbb Z _ {60}$ is abelian, this is automatically a principal series.
    • F: Example: For $n \ge 5$, $S _ n \supset A _ n \supset {(1)}$ is a composition series for $S _ n$ as $S _ n / A _ n \cong \mathbb Z _ 2$ and $A _ n$ is simple.
    • Example: Not every group has a principal series. Suppose $\mathbb Z = H _ n \supset H _ {n-1} \supset \cdots \supset H _ 1 \supset H _ 0 = {0}$ is a subnormal series for the integers under addition. Then $H _ 1$ is of the form $k \mathbb Z$ for $k \in \mathbb N$. But then $H _ 1 / H _ 0$ is an infinite cyclic group with many nontrivial proper normal subgroups
    • F: Theorem: Jordan-Hölder: Any two composition series of $G$ are isomorphic.
    • F: Definition: A group $G$ is solvable if it has a subnormal series ${H _ i}$ such that all the factor groups $H _ {i+1}/H _ i$ are abelian.
    • Example:
      • F: $S _ 4$ is solvable since $S _ 4 \supset A _ 4 \supset {(1), (12)(34), (13)(24), (14)(23)} \supset {(1)}$ has abelian factor groups
      • F: $S _ 5$ is not solvable since for $n \ge 5$, $S _ n \supset A _ n \supset {(1)}$ is a composition series for $S _ n$ but $A _ n$ is not abelian

Chapter 21: Fields

  • Extension Fields
    • Definition of an extension field and a base field
    • Example of $\mathbb Q[\sqrt{2} + \sqrt{3}]$ being extension field of $\mathbb Q[\sqrt 2]$
    • Example of extension of $\mathbb Z _ 2$ to add a root of $1 + x + x^2$
    • “Fundamental theorem of Field Theory”: For any nonconstant polynomial over a field $F$, there exists an extension field $E$ such that there’s some $a \in E$ where $p(\alpha) = 0$.
    • Example of another extension of $\mathbb Z _ 2$ to add roots for $1 + x^4 + x^5$
    • Definition of algebraic element: $\alpha$ is algebraic over $F$ if you can find a polynomial over $F$ that annihiliates
    • Definition of transcendental: not algebraic
    • Definition of algebraic extension $E / F$: Every element of $E$ is algebraic over $F$
    • Definition of $F(\alpha _ 1, \ldots, \alpha _ n)$ as smallest field containing all $\alpha _ i$
    • F: Definition of $E/F$ being a simple extension of $F$: $E = F(\alpha)$ for some $\alpha$
    • Example that $\pi$ and $e$ are algebraic over the reals, but nontrivial to show they are transcendental over the rationals
    • Definition of an “algebraic number”: algebraic over $\mathbb Q$
    • Definition of “transcendental”: not algebraic over $\mathbb Q$
    • Example showing $\sqrt{2 + \sqrt 3}$ is algebraic over $\mathbb Q$
    • Theorem: If $E/F$ is a field extension and $\alpha \in E$, then $\alpha$ is transcendental over $F$ iff $F(\alpha)$ is isomorphic to $F(x)$, the field of fractions for $F[x]$.
    • Theorem: existence of a unique minimal polynomial for arbitrary extension fields
    • Example: $x^2 - 2$ is the minimal polynomial of $\sqrt 2$
    • Proposition: $F(\alpha) \cong F[x]/\langle m _ \alpha(x) \rangle$
    • Theorem: In a simple extension $F(\alpha) : F$ where $\alpha$ is algebraic, every element is a linear combination of powers of $\alpha$
    • Example: construction of real numbers as $\mathbb R[x] / \langle x^2 + 1\rangle$
    • Definition of a finite extension: $[E : F] = n$
    • Theorem: every finite extension $E/F$ is an algebraic extension
    • Remark: Converse is not true, i.e. not every algebraic extension is a finite extension (take algebraic completion of the rationals)
    • Theorem: tower law
    • Corollary: you can find the degree of a sequence of field extensions by multiplying up the intermediate ones
    • Corollary: Suppose $E/F$ is an extension and $\alpha \in E$ is algebraic with minimal polynomial $m _ \alpha$. If $\beta \in F(\alpha)$ and has minimal polynomial $m _ \beta$, then $\deg m _ \beta \mid \deg m _ \alpha$.
    • Example: $\mathbb Q(\sqrt 3, \sqrt 5) = \mathbb Q(\sqrt 3 + \sqrt 5)$
    • Example: $\mathbb Q(\sqrt[6]{5} i) = \mathbb Q(\sqrt[3]{5}, \sqrt 5 i)$
    • Theorem: several equivalent statements
      • $E$ is a finite extension of $F$
      • There exists a finite number of algebraic elements $\alpha _ 1, \ldots, \alpha _ n \in E$ such that $E = F(\alpha _ 1, \ldots, \alpha _ n)$.
      • There exists a sequence of fields $E = F(\alpha _ 1, \ldots, \alpha _ n) \supset F(\alpha _ 1, \ldots, \alpha _ {n-1}) \supset \cdots \supset F(\alpha _ 1) \supset F$
    • Theorem: If $E/F$, then the elements of $E$ that are algebraic over $F$ form a field
    • Corollary: The set of all algebraic numbers forms a field
    • Definition: Algebraic closure of a field $F$ contains all elements that are algebraic over $F$
    • Theorem: $F$ is algebraically closed iff every nonconstant polynomial in $F$ factors into linear factors over $F[x]$
    • Definition: A field $F$ is algebraically closed if every nonconstant polynomial over $F$ has a root in $F$
    • Corollary: An algebraically closed field $F$ has no proper algebraic extension $E$
    • Theorem: Every $F$ has a unique algebraic closure (difficult to prove)
  • Splitting Fields
    • Definition: $E/F$ is a splitting field of $p(x)$ if there exist elements $\alpha _ 1, \ldots, \alpha _ n$ in $E$ such that $E = F(\alpha _ 1, \ldots, \alpha _ n)$ and $p(x) = (x - \alpha _ 1) \cdots (x - \alpha _ n)$.
    • Definition: A polynomial $p(x) \in F[x]$ splits in $E$ if it is the product of linear factors in $E[x]$.
    • Example: $p(x) = x^4 + 2x^2 - 8$ has splitting field $\mathbb Q[\sqrt 2, i]$
    • Example: $p(x) = x^3 - 2$ has a root in the field $\mathbb Q[\sqrt[3]{3}]$, but this is not a splitting field since it doesn’t contain the complex roots
    • F: Theorem: Every $p(x) \in F[x]$ nonconstant has a splitting field $E$
    • Lemma: If $\phi : E \to F$ is a field isomorphism, $K$ is an extension field of $E$, $\alpha \in K$ is algebraic over $E$ with minimal polynomial $m _ \alpha$, $L$ is an extension field of $F$ such that $\beta$ is the root of the polynomial in $F[x]$ obtained from $p(x)$ under the image of $\phi$. Then $\phi$ extends to a unique isomorphism $\overline \phi : E(\alpha) \to F(\beta)$ such that $\overline \phi(\alpha) = \beta$ and $\overline \phi$ agrees with $\phi$ on $E$
    • Theorem: Field extensions are unique, sort of: Let $\phi : E \to F$ be an isomorphism of fields, $p(x)$ is a nonconstant polynomial in $E[x]$ and $q(x)$ is corresponding polynomial in $F[x]$ under the isomoprhism. If $k$ is a splitting field of $p(x)$ and $L$ is a splitting field of $q(x)$, then $\phi$ extends to an isomorphism $\psi : K \to L$.
  • Geometric Constructions
  • Reading Questions
  • Exercises
  • References and Suggested Readings
  • Sage
  • Sage Exercises

Chapter 22: Finite Fields

  • Structure of a Finite Field
    • Proposition: Finite fields have prime characteristic
    • Proposition: If $F$ is a finite field of characteristic $p$, then order of $F$ is $p^n$ for some $n \in \mathbb N$.
    • Lemma: Freshman’s dream
    • Definition: A polynomial $f(x) \in F[x]$ of degree $n$ is separable if it has $n$ distnct roots in splitting field of $f(x)$
    • Definition: An extension $E$ of $F$ is separable extension of $F$ if every element in $E$ is the root of a separable polynomial in $F[x]$
    • Example: $x^2 - 2$ is separable over $\mathbb Q$
    • Example: $\mathbb Q[\sqrt 2]$ is a separable extension of $\mathbb Q$ (check every element is the root of a separable polynomial)
    • Definition: Derivative of a polynomial
    • F: Lemma: If $F$ is a field and $f(x) \in F[x]$, then $f$ is a separable iff $f(x)$ and $f’(x)$ are relatively prime
    • F: Theorem: For every prime $p$ and every positive integer $n$, there exists a finite field with $p^n$ elements. Furthermore, any field of order $p^n$ is isomorphic to the splitting field of $x^{p^n} - x$ over $\mathbb Z _ p$
    • Definition: The (unique up to isomorphism) field with $p^n$ elements is called the Galois field of order $p^n$, denoted $\text{GF}(p^n)$.
    • Theorem: Every subfield of $\text{GF}(p^n)$ has $p^m$ elements where $m \mid n$, and if $m \mid n$ for $m > 0$, there exists a unique subfield of $GF(p^n)$ isomorphic to $\text{GF}(p^m)$.
    • Example: subfields of $\text{GF}(p^{24})$
    • Theorem: If $F$ is a field and $F^\ast$ is the multiplicative group of nonzero elements then if $G$ is a finite subgroup of $F^\ast$, $G$ is cyclic
    • Corollary: Multiplicative group of all nonzero elements of a finite field is cyclic
    • Corollary: Every finite extension $E$ of a finite field $F$ is a simple extension of $F$
    • Example: $\text{GF}(2^4)$ is isomorphic $\mathbb Z _ 2 / \langle 1 + x + x^4 \rangle$
  • Polynomial Codes
  • Reading Questions
  • Exercises
  • Additional Exercises: Error Correction for BCH Codes
  • References and Suggested Readings
  • Sage
  • Sage Exercises

Chapter 23: Galois Theory

  • Field Automorphisms
    • Proposition: The set of all automorphisms of a field $F$ is a group under composition.
    • Proposition: The set of all $F$-linear automorphisms of a field extension $E/F$ of a group under composition.
    • (the $F$-linear automorphisms of a field extension $E/F$ are the automorphisms of $E$ which fix $F$ elementwise)
    • Definition: If $E/F$ is a field extension, define $\text{Gal}(E/F)$ as the set of all $F$-linear automorphisms of the field
    • Definition: If $f(x)$ is a polynomial in $F[x]$ and $E$ is the splitting field of $f(x)$ over $F$, then the Galois group of $f$ is defined as $\text{Gal}(E/F)$.
    • Example: complex conjugation is in $\text{Gal}(\mathbb C / \mathbb R)$.
    • Example: $\mathbb Q$-linear automorphisms of $\mathbb Q[\sqrt 3, \sqrt 5]$ give a group of size $4$
    • F: Proposition: If $E/F$ is a field of extension of $f(x) \in F[x]$, then any element of $\text{Gal}(E/F)$ defines a permutation of the roots of $f$ that lie in $E$
    • F: Proposition: Any permutation of the roots of $f$ that lie in $E$ is also an element of $\text{Gal}(E/F)$
    • Definition: If $E/F$ is an algebraic extension, two elements $\alpha, \beta \in E$ are said to be conjugate if they have the same minimal polynomial.
    • Example: $\sqrt 2$ and $-\sqrt 2$ are conjugate over $\mathbb Q$ since they’re both roots of $x^2 - 2$.
    • Proposition: If $\alpha$ and $\beta$ are conjugate over $F$, then there exists an isomorphism $\sigma : F(\alpha) \to F(\beta)$ such that $\sigma$ is the identity when restricted to $F$
    • F: Theorem: Let $f(x)$ be a polynomial in $F[x]$ and suppose that $E$ is the splitting field for $f(x)$ over $F$. If $f(x)$ has no repeated roots, then $ \vert \text{Gal}(E/F) \vert = [E : F]$.
    • F: Corollary: Let $F$ be a finite field with a finite extension $E$ such that $[E : F] = k$. Then $\text{Gal}(E/F)$ is cyclic of order $k$.
    • Example: $\text{Gal}(\mathbb Q[\sqrt 3, \sqrt 5] / \mathbb Q) \cong \mathbb Z _ 2 \times \mathbb Z _ 2$
    • Example: $\text{Gal}(\mathbb Q[\omega] : \mathbb Q) = \mathbb Z _ 4$ where $\omega$ is a root of $f(x) = x^4 + x^3 + x^2 + x + 1$.
    • Definition: Multiplicity of a root
    • Definition: Simple root is a root with multiplicity 1
    • Proposition: Let $f(x)$ be an irreducible polynomial over $F$. If the characteristic of $F$ is $0$, then $f(x)$ is separable. If the characteristic of $F$ is $p$ and $f(x) \ne g(x^p)$ for some $g(x)$ in $F[x]$, then $f(x)$ is also separable
    • Definition: $\alpha \in E$ is a primitive element of the field extension $E/F$ if $E = F(\alpha)$.
    • F: Theorem: Primitive element theorem: Let $E$ be a finite separable extension of a field $F$, then there exists an $\alpha \in E$ such that $E = F(\alpha)$.
  • The Fundamental Theorem
    • Proposition: Let $S = {\sigma _ i \mid i \in I}$ be a collection of automorphisms of $F$. Then $F^{S} = {a \in F \mid \sigma _ i(a) = a \text{ for all } \sigma _ i \in S}$ is a subfield of $F$
    • Corollary: If $F$ is a field and $G$ is a subgroup of $\text{Aut}(G)$, then $F^G = {\alpha \in F \mid \sigma(a) = a \text{ for all } \sigma \in G}$ is a subfield of $F$.
    • Definition: The subfield $F^S$ is called the fixed field of $S$.
    • Example: If $\sigma : \mathbb Q(\sqrt 3, \sqrt 5) \to \mathbb Q(\sqrt 3, \sqrt 5)$ is the automorphism that maps $\sqrt 3$ to $-\sqrt 3$, then $\mathbb Q(\sqrt 3, \sqrt 5)^{\langle\sigma\rangle} = \mathbb Q[\sqrt 5]$, i.e. the subfield left alone
    • Proposition: Let $E$ be a splitting field of $F$ of a separable polynomial. Then $E^{\text{Gal}(E/F)} = F$.
    • Lemma: $E$ is a field. Let $G$ be a finite group of automorphisms of $E$ and let $F = E^G$. Then $[E : F] \le \vert G \vert $. (Why important?)
    • Definition: If $E/F$ is an algebraic extension and every irreducible polynomial in $F[x]$ with a root in $E$ has all of its roots in $E$, then $E$ is called a normal extension of $F$. In other words, every irreducible polynomial in $F[x]$ containing a root in $E$ is the product of linear factors in $E[x]$.
    • Theorem: $E/F$ is a finite extension. The following are equivalent:
      • $E$ is a finite, normal, separable extension
      • $E$ is a splitting field over $F$ of a separable polynomial
      • $F = E^G$ for some finite group $G$ of automorphisms of $E$
    • Corollary: If $K/F$ is a field extension such that $F = K^G$ for some finite group of automorphisms $G$ of $K$, then $G = \text{Gal}(K/F)$.
    • Example: Lattice of subgroups of $\text{Gal}(\mathbb Q(\sqrt 3, \sqrt 5)/\mathbb Q)$ and the lattice of subfields of $\mathbb Q(\sqrt 3, \sqrt 5)$.
    • Theorem: Fundamental Theorem of Galois Theory: Let $F$ be a finite field or a field of characteristic zero. If $E$ is a finite normal extension of $F$ with Galois group $\text{Gal}(E/F)$, then the following holds:
      • The map $K \mapsto \text{Gal}(E/K)$ is a bijection of subfields $K$ of $E$ containing $F$ with the subgroups of $\text{Gal}(E/F)$.
      • If $F \subset K \subset E$, then $[E : K] = \vert G(E/K) \vert $ and $[K : F] = [G(E/F) : G(E/K)]$
      • $F \subset K \subset L \subset E$ iff ${\text{id}} \subset \text{Gal}(E/L) \subset \text{Gal}(E/K) \subset \text{Gal}(E/F)$.
      • $K$ is a normal extension of $F$ iff $\text{Gal}(E/K)$ is a normal subgroup of $\text{Gal}(E/F)$. In this case, $\text{Gal}(K/F) \cong \text{Gal}(E/F) / \text{Gal}(E/K)$.
    • Example: Lattice of subgroups of Galois group of $f(x) = x^4 - 2$ compared to the lattice of field extensions of the splitting field of $x^4 - 2$.
  • Applications
    • Definition: An extension $E/F$ is called an extension by radicals if there is a chain of subfields $F = F _ 0 \subset F _ 1 \cdots F _ r = E$ for $i = 1, 2, \ldots, r$ and $F _ i = F _ {i-1}(\alpha _ i)$ and $\alpha _ i^n \in F _ {i-1}$.
    • Definition: A polynomial $f(x)$ is solvable by radicals over $F$ if the splitting field $K$ of $f(x)$ over $F$ is contained in an extension of $F$ by radicals
    • Example: $x^n - 1$ is solvable by radicals over $\mathbb Q$ for any $n$
    • Lemma: Let $F$ be a field of characteristic zero and $E$ be the splitting field of $x^n - a$ over $F$ with $a \in F$. Then $\text{Gal}(E/F)$ is a solvable group.
    • Lemma: Let $F$ be a field of characteristic zero and let $F = F _ 0 \subset F _ 1 \subset \cdots \subset F _ r = E$ be a radical extension of $F$. Then there exists a normal radical extension $F = K _ 0 \subset K _ 1 \subset \cdots \subset K _ r = K$ such that $K$ contains $E$ and $K _ i$ and $K _ i$ is a normal extension of $K _ {i-1}$.
    • Theorem: Let $f(x)$ be in $F[x]$ where the characteristic of $F$ is zero. Then $f(x)$ is solvable by radicals iff the Galois group of $f(x)$ over $F$ is solvable.
    • Lemma: If $p$ is a prime, then any subgroup of $S _ p$ that contains a transposition and a cycle of length $p$ must be all of $S _ p$.
    • Example: $f(x) = x^5 - 6x^3 - 27x - 3 \in \mathbb Q[x]$ has Galois group $S _ 5$. But $S _ 5$ is not solvable, so this polynomial is not solvable by radicals.
    • Theorem: Fundamental Theorem of Algebra: The field of complex numbers is algebraically closed, i.e. every polynomial $\mathbb C[x]$ has a root in $\mathbb C$.
  • Reading Questions
  • Exercises
  • References and Suggested Readings
  • Sage
  • Sage Exercises


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