Rings, Modules, Fields
\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\require {mathtools}\)
\(\newenvironment {crampedsubarray}[1]{}{}\)
\(\newcommand {\smashoperator }[2][]{#2\limits }\)
\(\newcommand {\SwapAboveDisplaySkip }{}\)
\(\newcommand {\LaTeXunderbrace }[1]{\underbrace {#1}}\)
\(\newcommand {\LaTeXoverbrace }[1]{\overbrace {#1}}\)
\(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\)
\(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\)
\(\let \LWRorigshoveleft \shoveleft \)
\(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\)
\(\let \LWRorigshoveright \shoveright \)
\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)
\(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\vcentcolon }{\mathrel {\unicode {x2236}}}\)
\(\def \LWRtensorindicesthreesub #1#2{{_{#2}}\LWRtensorindicesthree }\)
\(\def \LWRtensorindicesthreesup #1#2{{^{#2}}\LWRtensorindicesthree }\)
\(\newcommand {\LWRtensorindicesthreenotsup }{}\)
\(\newcommand {\LWRtensorindicesthreenotsub }{ \ifnextchar ^ \LWRtensorindicesthreesup \LWRtensorindicesthreenotsup }\)
\(\newcommand {\LWRtensorindicesthree }{ \ifnextchar _ \LWRtensorindicesthreesub \LWRtensorindicesthreenotsub }\)
\(\newcommand {\LWRtensorindicestwo }{ \ifstar \LWRtensorindicesthree \LWRtensorindicesthree }\)
\(\newcommand {\indices }[1]{\LWRtensorindicestwo #1}\)
\(\newcommand {\LWRtensortwo }[3][]{{}\indices {#1}{#2}\indices {#3}}\)
\(\newcommand {\tensor }{\ifstar \LWRtensortwo \LWRtensortwo }\)
\(\newcommand {\LWRnuclidetwo }[2][]{{\vphantom {\mathrm {#2}}{}^{\LWRtensornucleonnumber }_{#1}\mathrm {#2}}}\)
\(\newcommand {\nuclide }[1][]{\def \LWRtensornucleonnumber {#1}\LWRnuclidetwo }\)
\(\newcommand {\FF }{\mathbb {F}}\)
\(\newcommand {\cO }{\mathcal {O}}\)
\(\newcommand {\cC }{\mathcal {C}}\)
\(\newcommand {\cP }{\mathcal {P}}\)
\(\newcommand {\cF }{\mathcal {F}}\)
\(\newcommand {\cS }{\mathcal {S}}\)
\(\newcommand {\cK }{\mathcal {K}}\)
\(\newcommand {\cM }{\mathcal {M}}\)
\(\newcommand {\GG }{\mathbb {G}}\)
\(\newcommand {\ZZ }{\mathbb {Z}}\)
\(\newcommand {\NN }{\mathbb {N}}\)
\(\newcommand {\PP }{\mathbb {P}}\)
\(\newcommand {\QQ }{\mathbb {Q}}\)
\(\newcommand {\RR }{\mathbb {R}}\)
\(\newcommand {\LL }{\mathbb {L}}\)
\(\newcommand {\HH }{\mathbb {H}}\)
\(\newcommand {\EE }{\mathbb {E}}\)
\(\newcommand {\SP }{\mathbb {S}}\)
\(\newcommand {\CC }{\mathbb {C}}\)
\(\newcommand {\FF }{\mathbb {F}}\)
\(\renewcommand {\AA }{\mathbb {A}}\)
\(\newcommand {\sF }{\mathscr {F}}\)
\(\newcommand {\sC }{\mathscr {C}}\)
\(\newcommand {\ts }{\textsuperscript }\)
\(\newcommand {\mf }{\mathfrak }\)
\(\newcommand {\cc }{\mf {c}}\)
\(\newcommand {\mg }{\mf {g}}\)
\(\newcommand {\ma }{\mf {a}}\)
\(\newcommand {\mh }{\mf {h}}\)
\(\newcommand {\mn }{\mf {n}}\)
\(\newcommand {\mc }{\mf {c}}\)
\(\newcommand {\ul }{\underline }\)
\(\newcommand {\mz }{\mf {z}}\)
\(\newcommand {\me }{\mf {e}}\)
\(\newcommand {\mff }{\mf {f}}\)
\(\newcommand {\mm }{\mf {m}}\)
\(\newcommand {\mt }{\mf {t}}\)
\(\newcommand {\pp }{\mf {p}}\)
\(\newcommand {\qq }{\mf {q}}\)
\(\newcommand {\gl }{\mf {gl}}\)
\(\newcommand {\msl }{\mf {sl}}\)
\(\newcommand {\so }{\mf {so}}\)
\(\newcommand {\mfu }{\mf {u}}\)
\(\newcommand {\su }{\mf {su}}\)
\(\newcommand {\msp }{\mf {sp}}\)
\(\renewcommand {\aa }{\mf {a}}\)
\(\newcommand {\bb }{\mf {b}}\)
\(\newcommand {\sR }{\mathscr {R}}\)
\(\newcommand {\lb }{\langle }\)
\(\newcommand {\rb }{\rangle }\)
\(\newcommand {\ff }{\mf {f}}\)
\(\newcommand {\ee }{\epsilon }\)
\(\newcommand {\heart }{\heartsuit }\)
\(\newcommand {\Mloc }{\mathcal {M}_{\text {loc}}}\)
\(\newcommand {\Mnilpnil }{\mathcal {M}_{\text {nil}}^{\text {pnil}}}\)
\(\newcommand {\Uloc }{\mathcal {U}_{\text {loc}}}\)
\(\newcommand {\Mnil }{\mathcal {M}_{\text {nil}}}\)
\(\newcommand {\Unil }{\mathcal {U}_{\text {nil}}}\)
\(\newcommand {\floor }[1]{\lfloor #1 \rfloor }\)
\(\newcommand {\ceil }[1]{\lceil #1 \rceil }\)
\(\newcommand {\pushout }{\arrow [ul, phantom, "\ulcorner ", very near start]}\)
\(\newcommand {\pullback }{\arrow [dr, phantom, "\lrcorner ", very near start]}\)
\(\newcommand {\simp }[1]{#1^{\Delta ^{op}}}\)
\(\newcommand {\arrowtcupp }[2]{\arrow [bend left=50, ""{name=U, below,inner sep=1}]{#1}\arrow [Rightarrow,from=U,to=MU,"#2"]}\)
\(\newcommand {\arrowtclow }[2]{\arrow [bend right=50, ""{name=L,inner sep=1}]{#1}\arrow [Rightarrow,from=LM,to=L]{}[]{#2}}\)
\(\newcommand {\arrowtcmid }[2]{\arrow [""{name=MU,inner sep=1},""{name=LM,below,inner sep=1}]{#1}[pos=.1]{#2}}\)
\(\newcommand {\dummy }{\textcolor {white}{\bullet }}\)
\(\newcommand {\adjunction }[4]{ #1\hspace {2pt}\colon #2 \leftrightharpoons #3 \hspace {2pt}\colon #4 }\)
\(\newcommand {\aug }{\mathop {\rm aug}\nolimits }\)
\(\newcommand {\MC }{\mathop {\rm MC}\nolimits }\)
\(\newcommand {\art }{\mathop {\rm art}\nolimits }\)
\(\newcommand {\DiGrph }{\mathop {\rm DiGrph}\nolimits }\)
\(\newcommand {\FMP }{\mathop {\rm FMP}\nolimits }\)
\(\newcommand {\CAlg }{\mathop {\rm CAlg}\nolimits }\)
\(\newcommand {\perf }{\mathop {\rm perf}\nolimits }\)
\(\newcommand {\cof }{\mathop {\rm cof}\nolimits }\)
\(\newcommand {\fib }{\mathop {\rm fib}\nolimits }\)
\(\newcommand {\Thick }{\mathop {\rm Thick}\nolimits }\)
\(\newcommand {\Orb }{\mathop {\rm Orb}\nolimits }\)
\(\newcommand {\ko }{\mathop {\rm ko}\nolimits }\)
\(\newcommand {\Spf }{\mathop {\rm Spf}\nolimits }\)
\(\newcommand {\Spc }{\mathop {\rm Spc}\nolimits }\)
\(\newcommand {\sk }{\mathop {\rm sk}\nolimits }\)
\(\newcommand {\cosk }{\mathop {\rm cosk}\nolimits }\)
\(\newcommand {\holim }{\mathop {\rm holim}\nolimits }\)
\(\newcommand {\hocolim }{\mathop {\rm hocolim}\nolimits }\)
\(\newcommand {\Pre }{\mathop {\rm Pre}\nolimits }\)
\(\newcommand {\THR }{\mathop {\rm THR}\nolimits }\)
\(\newcommand {\THH }{\mathop {\rm THH}\nolimits }\)
\(\newcommand {\Fun }{\mathop {\rm Fun}\nolimits }\)
\(\newcommand {\Loc }{\mathop {\rm Loc}\nolimits }\)
\(\newcommand {\Bord }{\mathop {\rm Bord}\nolimits }\)
\(\newcommand {\Cob }{\mathop {\rm Cob}\nolimits }\)
\(\newcommand {\Set }{\mathop {\rm Set}\nolimits }\)
\(\newcommand {\Ind }{\mathop {\rm Ind}\nolimits }\)
\(\newcommand {\Sind }{\mathop {\rm Sind}\nolimits }\)
\(\newcommand {\Ext }{\mathop {\rm Ext}\nolimits }\)
\(\newcommand {\sd }{\mathop {\rm sd}\nolimits }\)
\(\newcommand {\Ex }{\mathop {\rm Ex}\nolimits }\)
\(\newcommand {\Out }{\mathop {\rm Out}\nolimits }\)
\(\newcommand {\Cyl }{\mathop {\rm Cyl}\nolimits }\)
\(\newcommand {\Path }{\mathop {\rm Path}\nolimits }\)
\(\newcommand {\Ch }{\mathop {\rm Ch}\nolimits }\)
\(\newcommand {\SSet }{\mathop {\rm \Set ^{\Delta ^{op}}}\nolimits }\)
\(\newcommand {\Sq }{\mathop {\rm Sq}\nolimits }\)
\(\newcommand {\Free }{\mathop {\rm Free}\nolimits }\)
\(\newcommand {\Map }{\mathop {\rm Map}\nolimits }\)
\(\newcommand {\Chain }{\mathop {\rm Ch}\nolimits }\)
\(\newcommand {\LMap }{\mathop {\rm LMap}\nolimits }\)
\(\newcommand {\RMap }{\mathop {\rm RMap}\nolimits }\)
\(\newcommand {\Tot }{\mathop {\rm Tot}\nolimits }\)
\(\newcommand {\MU }{\mathop {\rm MU}\nolimits }\)
\(\newcommand {\MSU }{\mathop {\rm MSU}\nolimits }\)
\(\newcommand {\MSp }{\mathop {\rm MSp}\nolimits }\)
\(\newcommand {\MSO }{\mathop {\rm MSO}\nolimits }\)
\(\newcommand {\MO }{\mathop {\rm MO}\nolimits }\)
\(\newcommand {\BU }{\mathop {\rm BU}\nolimits }\)
\(\newcommand {\KU }{\mathop {\rm KU}\nolimits }\)
\(\newcommand {\BSU }{\mathop {\rm BSU}\nolimits }\)
\(\newcommand {\BSp }{\mathop {\rm BSp}\nolimits }\)
\(\newcommand {\BGL }{\mathop {\rm BGL}\nolimits }\)
\(\newcommand {\BSO }{\mathop {\rm BSO}\nolimits }\)
\(\newcommand {\BO }{\mathop {\rm BO}\nolimits }\)
\(\newcommand {\KO }{\mathop {\rm KO}\nolimits }\)
\(\newcommand {\Tor }{\mathop {\rm Tor}\nolimits }\)
\(\newcommand {\Cotor }{\mathop {\rm Cotor}\nolimits }\)
\(\newcommand {\imag }{\mathop {\rm Im}\nolimits }\)
\(\newcommand {\real }{\mathop {\rm Re}\nolimits }\)
\(\newcommand {\Cat }{\mathop {\rm Cat}\nolimits }\)
\(\newcommand {\Fld }{\mathop {\rm Fld}\nolimits }\)
\(\newcommand {\Frac }{\mathop {\rm Frac}\nolimits }\)
\(\newcommand {\Dom }{\mathop {\rm Dom}\nolimits }\)
\(\newcommand {\Hotc }{\mathop {\rm Hotc}\nolimits }\)
\(\newcommand {\Top }{\mathop {\rm Top}\nolimits }\)
\(\newcommand {\Ring }{\mathop {\rm Ring}\nolimits }\)
\(\newcommand {\CRing }{\mathop {\rm CRing}\nolimits }\)
\(\newcommand {\CGHaus }{\mathop {\rm CGHaus}\nolimits }\)
\(\newcommand {\Alg }{\mathop {\rm Alg}\nolimits }\)
\(\newcommand {\Bool }{\mathop {\rm Bool}\nolimits }\)
\(\newcommand {\hTop }{\mathop {\rm hTop}\nolimits }\)
\(\newcommand {\Nat }{\mathop {\rm Nat}\nolimits }\)
\(\newcommand {\Rel }{\mathop {\rm Rel}\nolimits }\)
\(\newcommand {\Mod }{\mathop {\rm Mod}\nolimits }\)
\(\newcommand {\Space }{\mathop {\rm Space}\nolimits }\)
\(\newcommand {\Vect }{\mathop {\rm Vect}\nolimits }\)
\(\newcommand {\FinVect }{\mathop {\rm FinVect}\nolimits }\)
\(\newcommand {\Matr }{\mathop {\rm Matr}\nolimits }\)
\(\newcommand {\Ab }{\mathop {\rm Ab}\nolimits }\)
\(\newcommand {\Gr }{\mathop {\rm Gr}\nolimits }\)
\(\newcommand {\Grp }{\mathop {\rm Grp}\nolimits }\)
\(\newcommand {\Hol }{\mathop {\rm Hol}\nolimits }\)
\(\newcommand {\Gpd }{\mathop {\rm Gpd}\nolimits }\)
\(\newcommand {\Grpd }{\mathop {\rm Gpd}\nolimits }\)
\(\newcommand {\Mon }{\mathop {\rm Mon}\nolimits }\)
\(\newcommand {\FinSet }{\mathop {\rm FinSet}\nolimits }\)
\(\newcommand {\Sch }{\mathop {\rm Sch}\nolimits }\)
\(\newcommand {\AffSch }{\mathop {\rm AffSch}\nolimits }\)
\(\newcommand {\Idem }{\mathop {\rm Idem}\nolimits }\)
\(\newcommand {\SIdem }{\mathop {\rm SIdem}\nolimits }\)
\(\newcommand {\Aut }{\mathop {\rm Aut}\nolimits }\)
\(\newcommand {\Ord }{\mathop {\rm Ord}\nolimits }\)
\(\newcommand {\coker }{\mathop {\rm coker}\nolimits }\)
\(\newcommand {\ch }{\mathop {\rm char}\nolimits }\)
\(\newcommand {\Sym }{\mathop {\rm Sym}\nolimits }\)
\(\newcommand {\adj }{\mathop {\rm adj}\nolimits }\)
\(\newcommand {\dil }{\mathop {\rm dil}\nolimits }\)
\(\newcommand {\Cl }{\mathop {\rm Cl}\nolimits }\)
\(\newcommand {\Diff }{\mathop {\rm Diff}\nolimits }\)
\(\newcommand {\End }{\mathop {\rm End}\nolimits }\)
\(\newcommand {\Hom }{\mathop {\rm Hom}\nolimits }\)
\(\newcommand {\Gal }{\mathop {\rm Gal}\nolimits }\)
\(\newcommand {\Pos }{\mathop {\rm Pos}\nolimits }\)
\(\newcommand {\Ad }{\mathop {\rm Ad}\nolimits }\)
\(\newcommand {\GL }{\mathop {\rm GL}\nolimits }\)
\(\newcommand {\SL }{\mathop {\rm SL}\nolimits }\)
\(\newcommand {\vol }{\mathop {\rm vol}\nolimits }\)
\(\newcommand {\reg }{\mathop {\rm reg}\nolimits }\)
\(\newcommand {\Or }{\textnormal {O}}\)
\(\newcommand {\U }{\mathop {\rm U}\nolimits }\)
\(\newcommand {\SOr }{\mathop {\rm SO}\nolimits }\)
\(\newcommand {\SU }{\mathop {\rm SU}\nolimits }\)
\(\newcommand {\Spin }{\mathop {\rm Spin}\nolimits }\)
\(\newcommand {\Sp }{\mathop {\rm Sp}\nolimits }\)
\(\newcommand {\Int }{\mathop {\rm Int}\nolimits }\)
\(\newcommand {\im }{\mathop {\rm im}\nolimits }\)
\(\newcommand {\dom }{\mathop {\rm dom}\nolimits }\)
\(\newcommand {\di }{\mathop {\rm div}\nolimits }\)
\(\newcommand {\cod }{\mathop {\rm cod}\nolimits }\)
\(\newcommand {\colim }{\mathop {\rm colim}\nolimits }\)
\(\newcommand {\ad }{\mathop {\rm ad}\nolimits }\)
\(\newcommand {\PSL }{\mathop {\rm PSL}\nolimits }\)
\(\newcommand {\PGL }{\mathop {\rm PGL}\nolimits }\)
\(\newcommand {\sep }{\mathop {\rm sep}\nolimits }\)
\(\newcommand {\MCG }{\mathop {\rm MCG}\nolimits }\)
\(\newcommand {\oMCG }{\mathop {\rm MCG^+}\nolimits }\)
\(\newcommand {\Spec }{\mathop {\rm Spec}\nolimits }\)
\(\newcommand {\rank }{\mathop {\rm rank}\nolimits }\)
\(\newcommand {\diverg }{\mathop {\rm div}\nolimits }\)
\(\newcommand {\disc }{\mathop {\rm disc}\nolimits }\)
\(\newcommand {\sign }{\mathop {\rm sign}\nolimits }\)
\(\newcommand {\Arf }{\mathop {\rm Arf}\nolimits }\)
\(\newcommand {\Pic }{\mathop {\rm Pic}\nolimits }\)
\(\newcommand {\Tr }{\mathop {\rm Tr}\nolimits }\)
\(\newcommand {\res }{\mathop {\rm res}\nolimits }\)
\(\newcommand {\Proj }{\mathop {\rm Proj}\nolimits }\)
\(\newcommand {\mult }{\mathop {\rm mult}\nolimits }\)
\(\newcommand {\N }{\mathop {\rm N}\nolimits }\)
\(\newcommand {\lk }{\mathop {\rm lk}\nolimits }\)
\(\newcommand {\Pf }{\mathop {\rm Pf}\nolimits }\)
\(\newcommand {\sgn }{\mathop {\rm sgn}\nolimits }\)
\(\newcommand {\grad }{\mathop {\rm grad}\nolimits }\)
\(\newcommand {\lcm }{\mathop {\rm lcm}\nolimits }\)
\(\newcommand {\Ric }{\mathop {\rm Ric}\nolimits }\)
\(\newcommand {\Hess }{\mathop {\rm Hess}\nolimits }\)
\(\newcommand {\sn }{\mathop {\rm sn}\nolimits }\)
\(\newcommand {\cut }{\mathop {\rm cut}\nolimits }\)
\(\newcommand {\tr }{\mathop {\rm tr}\nolimits }\)
\(\newcommand {\codim }{\mathop {\rm codim}\nolimits }\)
\(\newcommand {\ind }{\mathop {\rm index}\nolimits }\)
\(\newcommand {\rad }{\mathop {\rm rad}\nolimits }\)
\(\newcommand {\Rep }{\mathop {\rm Rep}\nolimits }\)
\(\newcommand {\Lie }{\mathop {\rm Lie}\nolimits }\)
\(\newcommand {\Der }{\mathop {\rm Der}\nolimits }\)
\(\newcommand {\hgt }{\mathop {\rm ht}\nolimits }\)
\(\newcommand {\Ider }{\mathop {\rm Ider}\nolimits }\)
\(\newcommand {\id }{\mathop {\rm id}\nolimits }\)
11 Fields
Fields are simple objects in the category of rings. Since they are integral domains, their characteristic is either a prime \(p\) or \(0\). Much of the way we study fields relies on the fact that their module theory is very simple, in
particular, all modules are free. Moreover, any morphism of fields is injective since the kernels are proper ideals. Thus our basic notion is that of a field extension \(L/F\), which is just an inclusion \(F \hookrightarrow
L\). In particular, the field \(L\) becomes a \(F\)-vector space, so has a dimension which we call the degree of the extension, denoted \([L:F]\). Note that by Lemma 7.3 a finite degree extension (which we can just call finite) is algebraic. We will start by studying algebraic extensions, mostly finite ones.
We say that an extension \(L/K\) is simple if it is generated over \(K\) by one element, \(\alpha \), which we call the primitive element. The degree of \(\alpha \) is then \([L:K] = [K(\alpha
):K]\).
-
Proof. Just note that the span of \(1,\alpha ,\dots , \alpha ^{\deg (\alpha )-1}\) form a field containing \(a\), giving one inequality, and the other inequality comes from
applying the Cayley-Hamilton Theorem to \(\chi _L(\alpha )\). □
We already have enough to prove an interesting fact:
-
Proof. If \(K(t)/L/K\) are extensions, with \(L \neq K\), \(L\) contains some rational function \(f/g\), where \(f,g\) are relatively prime. \(g(t)f/g-f(t)\) then shows that \(K(t)\)
is a simple finite extension of \(L\), and its degree is \(\min _{f/g \in L}(\max (\deg f,\deg g))\). Then a minimizing \(f/g\) and using Lemma 11.1, \(L = K(f/g)\).
□
We use \(\bar {K}\) to denote the algebraic closure of \(K\). The algebraic closure behaves similarly to the “universal cover" of a space. In particular, we would like every algebraic extension to be describable in terms of an
algebraic closure.
-
Proof. Consider the poset of embeddings of subfields use Zorn’s Lemma to choose a maximal one, and note that we can always embed a simple extension, so that the maximal element is
everywhere defined. □
-
Proof. Given a field \(F\), write \(F = F_0\). Now inductively given \(F_i\) produce a field \(F_{i+1}\) by adjoining an element \(g_\alpha \) to \(F_i\) for each irreducible
polynomial \(f_\alpha \) over \(F_i\), and modding out by a maximal ideal containing \(f_\alpha (g_\alpha )\). Then the union of the \(F_i\) is algebraically closed. For uniqueness, given \(M,L\), two algebraic closures of
\(K\), By the embedding lemma \(L\) embeds into \(M\), but this embedding is an isomorphism as \(L\) is algebraically closed. □
Note that fields don’t have many limits, but if we consider the category of fields embedded in some fixed algebraic closure, then we get a lattice, which does have limits and colimits. The product of some fields \(F_i\) is then their
intersection, and the coproduct is called their compositum.