Field Theory
Field theory has been one of the main areas where I tried to understand algebra in a deeper and more connected way. I worked through the material mainly using Isaacs’ Algebra: A Graduate Course, covering polynomial rings, field extensions, splitting fields, separability, normality, finite fields, and Galois theory.
These notes are my attempt to organize the subject carefully and make the logical flow clear. I tried to go through each concept in depth, filling in proof details, adding relevant examples, and expanding arguments that are often presented too briefly.
Chapter 12: Rings, Ideal, and Modules
- Sym($\Omega$), End(A)
- Ring
- A abelian $\implies$ End(A) is a ring
- Subring
- Ring Homomorphism
- Ideals, e$\in$I$\implies$ I is a subring
- $1_R\in I\implies I=R$
- Principal Ideal, (a)
- Left/Right Annihilator of $X\subseteq R$
- Factor Ring, $R/I$
- $\pi:R\to R/I, \pi(r)=r+I$ is a ring homomorphism with $ker(\pi)=I$
- Unit, $U(R)$
- Division Ring
- Field
- Subfield Test
- If R commutative then: M max ideal of R$\iff$ R/M field
- R Module
Chapter 16: Polynomial Rings, PIDs, and UFDs
- Ring, Subring
- Integral Domain
- Principal Ideal Domain (PID)
- $R[X]$
- $R[X]$ is a ring, Unitary subring
- $R$ integral domain $\implies $ $R[X]$ is integral domain
- Division Algorithm
- $F$ is field $\implies$ $F[X]$ is a PID
- Irreducible element of $R$
- Wishes of a Domain
- Divisibility in a Domain
- Unique Factorization Domain (UFD)
- Prime Ideal
- Prime Element
- Irreducible, Prime, and Maximal Ideals in a PID
- R is a PID $\implies$ R is UFD
- If $R$ is UFD, then $a\in R$ irreducible $\iff$ $a$ is prime element
- Coprime/Relatively elements
- $R$ is PID $\implies$ $R$ is UFD
- $R$ is UFD $\implies $ $R[X]$ is UFD
- Field of Fraction, Frac($R$)
- Eisenstein criterion
- Frac(Z[i])=Q(i)
Chapter 17: Extension Fields
- Extension Fields
- Algebraic, Transcedental/F
- F(t)=Frac(F[t]) is a field
- Degree of field extension, $|E:F|=\dim_F(E)$
- If $R$ is a domain, $F\subseteq R$ then $|R:F|<\infty\implies R$ is a field
- Minimal Polynomial of an Algebraic Element over F
- $F[ \alpha]\cong F[X]/(m_{F,\alpha})$ and $I=(m_{F,\alpha})$ is the max ideal
- $|F[\alpha]:F|=\deg(m_{F,\alpha})=\dim_F(F[\alpha])$
- $\alpha$ algebraic/F $\iff$ $|F[\alpha]:F|<\infty\iff F[\alpha]=F(\alpha)$
- $\alpha$ transcedental/F $\implies F[\alpha]\cong F[X]$
- $F \subseteq E$ with $|E:F|<\infty\implies E$ algebraic extension
- $F \subseteq E \subseteq L\implies |L:F|=|L:E||E:F|$
- $A=\{\alpha\in E|\alpha\text{ is algebraic/F}\}\implies F\subseteq A\subseteq E$
- $F[\alpha,\beta]=(F[\alpha])[\beta]=(F[\beta])[\alpha]$
- $F\subseteq E\subseteq L$, E algebraic/F, L algebraic/E $\implies$ L algebraic/F
- $f\in F[X]\implies \exists E\supseteq F, \alpha\in E$ st $f(\alpha)=0$
- f splits/F
- Algebraically closed
- Algebraic closure
- Splitting field of f over F
- E is the sf of $f\in F[X]\implies |E:F|<\infty$
- Simple extension $E=F(\alpha)$, primitive element $\alpha$
- Finite Subgroups of $F^×$ are Cyclic
Chapter 18: Galois Theory
- Galois group of the extension
- $Fix\Big(Gal(E/F)\Big)\supseteq F$ $\&$ $Gal\Big(E/Fix(G)\Big)\supseteq G$
- $H_1\subseteq H_2\implies Fix(H_1)\supseteq Fix(H_2)$
- Extension of Field Isomorphisms to Simple Extensions
- Uniqueness of Splitting Fields
- Gal(E/F) permutes the roots of f$\in$F[X] in E
- Fix(Gal(E/F))=F $\implies$ E galois/F
- Normal Extension
- E Normal/F$\iff$ E is sf/F for some g$\in$F[X]
- E algebraic closure of F$\implies$E is normal/F
- $|E:F|=2\implies$E is normal/F
- Repeated root of f
- Separable Extension
- f$\in$F[X] separable/F if each irreducible factor of f in F[X] has distinct roots in a sf
- F$\subseteq$K$\subseteq$E, $\alpha\in$ E then $\alpha$ normal(separable)/F$\implies\alpha$ normal(separable)/K
- Char(E)=Char(F)
- Char(F)=0 or F finite$\implies$ Every alg extension is separable
- E galois extension of F$\iff$E is sf/F for some separable poly/F
- F$\subseteq$K$\subseteq$E, E galois/F$\implies$E galois/K
- Finite degree separable ext.$\implies$E=F[$\alpha$]
- E galois/F$\iff |Gal(E/F)|=|E:F|$
- G finite$\implies$ E galois/F $\&$ Gal(E/Fix(G))=G
- Fundamental Theorem of Galois Theorem (FTGT)
- Algebraic Closedness Criterion: Odd-Degree Polynomials and Square Roots
- Compositum of E and L, $\langle L,E\rangle$
- Natural Irrationalities
Chapter 19: Separability
- $\alpha$ repeated root $\iff$ $f(\alpha)=0$ $\&$ $f'(\alpha)=0$
- Derivation of R
- Der(R) abelian grp under $+$
- Lie Ring
- F-algebra
- F-derivation
- $\delta(uv)=u\delta(v)+\delta(u)v\forall u,v\in\mathcal{B}\implies\delta$ is a derivation
- f$\in$F[X] irreducible then f has distinct roots$\iff$f'$\neq 0$
- Inseparable Irreducible Polynomials have the Form $g(X^p)$
- $(a+b)^p=a^p+b^p$, $(ab)^p=a^pb^p\;\forall a,b\in$F
- Finite fields are perfect ($F^p=F$)
- f$\in$F[X] inseparable$\implies$char(F)=p $\&$ F not perfect$\implies$F infinite
- Char(F)=0 or F perfect of Char(F)=p$\implies$all f$\in$F[X] are separable
- Irreducible Polynomials in Characteristic \(p\): \(f(X)=g(X^{p^n})\) with \(g\) Separable
- Purely Inseparable Extensions
Chapter 20: Cyclotomy and Geometric Constructions
- $n^{th}$ root of unity
- $n^{th}$ root of unity in F form a cyclic subgroup of F$^×$
- Primitive $n^{th}$ root of unity
- $\exists$ Primitive $n^{th}$ root of unity in $E\supseteq$ F $\iff$Char(F) $\nmid$ n
- F has primitive $n^{th}$ root$\iff$ $x^n-1$ splits/F$\&$Char(F)$\nmid$n
- Primitive nth Roots and Splitting Field of $x^n-1$
- $n^{th}$ cyclotomic polynomial, $\Phi_n(X)$
- $n^{th}$ cyclotomic field, $Q_n=Q[\varepsilon_n]$
- Set of all units in $Z/nZ$ form a multiplicative group
- $Gal(Q_n/Q)\cong (Z/nZ)^×$ is abelian
- G abelian, o(x)=n, o(y)=m, gcd(m,n)=1$\implies$o(xy)=mn
- gcd(m,n)=1$\implies Q_n\cap Q_m=Q$
- Kummer Theorem
- Crossed Homomorphism
- Dedekind's Lemma
- Constructible Numbers
- $K\subseteq \mathbb{C}$
- $\alpha\in K\implies\sqrt{\alpha}\in K$
- Constructible Elements and Towers of Quadratic Extensions
- $\alpha\in K\implies \alpha\text{algebraic/Q}\&|Q(\alpha):Q|=2^k$
- $\alpha\in K\iff \alpha\text{algebraic/Q}\&|Gal(E/Q)|=|E:Q|=2^k$
- Fermat prime
- regular n-gon is constructible $\iff \varepsilon_n\in$K
Chapter 21: Finite Fields
- $K=\cap$(intersection of subfields of F)
- F finite field$\&$|E:F|=n<$\infty\implies$E is finite$\&|E|=|F|^n$
- F finite field$\implies$Char(F)=p$\&$|F|=p^n where $|F:K|=n$
- $|F|^n\&F\supseteq K$ be prime subfield$\implies$every $\alpha\in F$ is a root of f
- $|E|=|F|<\infty\implies E\cong F$
- $GF(p^n)=\mathbb{F}_{p^n}$
- $E=GF(p^n)\implies E$ galois/K$\&$Gal(E/K)=<$\sigma$> where $|Gal(E/K)|=n$
- $|E|=p^n$ then $\exists$ ! $F\subseteq E$ with $|F|=p^m$ iff $m|n$