Vol. 4. No. 1, 41-45, Aril 2001, ISSN : 1410-8518 KETERHUBUNGAN GALOIS FIELD DAN LAPANGAN PEMISAH Bmbg Irwto Jurus Mtemtik FMIPA UNDIP Abstct I this er, it ws lered of the ecessry d sufficiet coditio for fiite field with elemets, rime d 1 iteger. A field F is etetio field of field K if K subfield F. The etesio field F of field K is Slittig field of collectio oliomil { f i () i I } of K if F smllest subfield K cotiig K d ll the zeros i K of the oliomil f i (). The zeros of oliomil f i () re elemets of field F d the elemets of F is fiite the F is fiite field (Glois fileld). F is fiite with elemets, rime d 1 iteger if oly if F is Slittig field of - over Z. Keywords : etetio fields, slittig fields, fiite fields. 1. PENDAHULUAN Lg dlh derh itegrl yg seti eleme yg tidk ol memuyi ivers terhd ergd. Lg disebut lg berhigg jik lg memiliki jumlh eleme yg berhigg. Lg berhigg serig jug disebut deg Glois Field (Risighi, M.D, 1980). Slh stu motivsi yg meltrbelkgi egerti dri Glois Field yitu lg erlus F ts lg K d K dlh sub field dri F (Hugerford, T. W, 1984). Mislk lg K, oliomil f () K[] d K dlh kr dri f () jk d hy jik ( ) fktor dri f () (Hugerford, 1984). Dlm tulis ii dieljri hubug Glois Field deg lg emish. 41
Keterhubug Glois Field (Bmbg Irwto) 2. LAPANGAN PEMISAH Lg F disebut lg erlus ts lg K jik K subfield dri F (Hugerford,T. W, 1984). Misl lg F deg krkteristik (rim) mk F memut sub field yg isomorfis deg Z, jik krkteristik F sm deg 0 mk F memut sub field yg isimorfis deg Q (himu bilg rsiol). Jik oliomil tk ol f () K [] d α F sedemiki higg f (α) = 0 mk α disebut eleme ljbr sebliky disebut trsedetl (Frleigh, J. B, 1994). Defiisi 1. F lg erlus ts lg K. Misl himu K F = {α F / α ljbr ts K} disebut eutu ljbr (ljbric closure). Defiisi 2. Misl K sutu lg deg eutu ljbr (lgebric closure) K. { f i () / i I } koleksi dri oliomil-oliomil dlm K[]. Sutu lg F K disebut lg emish (slittig field) dri { f i () / i I }ts K jik F dlh sub field terkecil dri K yg memut K d semu kr dlm K dri seti fi(), utuk i I. Sutu lg F K dlh lg emish (slittig field) ts K, jik F K dlh lg emish (slittig field) dri himu sebrg dri oliomil-oliomil dlm K[]. Semu lg K d semu f() K[] sedemiki higg deg (f) 1, terdtlh erlus F dri K yg meruk lg emish utuk (f) ts K (De R.A, 1996). 3. GALOIS FIELD Lg deg jumlh eleme berhigg disebut Glois Field. meulis bhw seti eleme dri lg berhigg K deg rim memeuhi ersyrt =. (Risighi, MD, 1980). Teorem 1. Misl F sutu lg deg eleme deg rim yg termut dlm eutu ljbr Z dri Z mk eleme-eleme F dlh krkr di Z dri oliomil Z[] } 42 - Z[] tu F = { Z / kr dri -
Vol. 4. No. 1, 41-45, Aril 2001, ISSN : 1410-8518 Bukti : Pdg himu F* yg meruk himu eleme-eleme yg tidk ol dlm F. Jels meruk F* gru multiliktif, deg order 1. Utuk α F*, order dri α dlm gru multiliktif membgi order 1 dri gru. Jdi utuk α F* dieroleh 1 α = 1, α α =. Sehigg utuk seti α F dlh kr dri ( - ) Z [] lig byk mk F memut tet krkr dri -, tu F = { Z / kr-kr dri ( - ) eleme Z [] }. Seljuty ditujukk bhw Z termut di dlm F, berrti utuk seti Z meruk kr dri ( deg iduksi Mtemtik. - ) tu utuk seti Z memeuhi (i) Ber utuk 1, sebb -1 = 1, jdi =. =. Bukti (ii) Jik ber utuk = k mk ber utuk = k + 1. Meurut hiotesis iduksi ber utuk = k, mk k =, seljuty (k +1) = k. = ( k ) = =. Jdi (k+1) = sehigg ber utuk = k + 1 jdi ber utuk seti, yg berrti terbukti bhw utuk seti Z memeuhi =. Berrti F dlh subfield dri Z terkecil yg memut Z d semu kr-kr dri ( - ) Z []. Jdi F lg emish dri - ts Z. Misl K sutu lg deg eutu ljbr (lgebric closure) K. {f i () / i I }koleksi dri oliomil-oliomil dlm K[]. Sutu lg F K disebut lg emish (slittig field) dri { f i () / i I } ts K jik F dlh sub field terkecil dri K yg memut K d semu kr dlm K dri seti f i (), utuk i I. Sutu lg F K dlh lg emish (slittig field) ts K, jik F K dlh lg emish (slittig field) dri himu sebrg dri oliomil-oliomil dlm K []. Semu lg K d semu f() K [] sedemiki sehigg deg (f) 1, terdtlh erlus F dri K yg meruk lg emish utuk f() 43
Keterhubug Glois Field (Bmbg Irwto) ts K (De R. A, 1996). Sedgk jik F sutu lg deg krkteristik rim 0 mk oliomil f() = - F[] utuk 1 memiliki krkr yg berbed. (Risighi,MD, 1980). Teorem 2. Misl rim d 1 dlh bilg bult, kr-kr oliomil - Z[] dlm lg emish ts Z yg semu berbed membetuk lg F deg eleme. Bukti : Misl oliomil f() = -, mk f 1 () = 1-1, kre rim d f() oliomil dlm lg emish Z jdi f 1 () = 0. 1-1 = -1 0 sehigg f() memiliki kr-kr yg berbed oliomil f() 1 - berderjt d memiliki kr-kr yg semu berbed sehigg jumlh kr-kry kr. Misl F himu semu kr-kr dri f() tu F = { Z kr dri f() = - } k ditujukk bhw F dlh lg. Ambil, b F mk - = 0 jdi = begitu jug b - b = 0 mk b = b, sehigg ( + b) - ( + b) = 0 dieroleh ( + b) = ( + b) seljuty utuk ( b -1 ) = (b ) 1 = b -1 Jdi F dlh lg, kre jumlh kry mk F dlh lg deg eleme tu GF ( ). 4. KESIMPULAN 44 F dlh lg deg eleme dim rim d 1 bilg bult bil d hy bil F meruk lg emish dri 5. UCAPAN TERIMA KASIH - ts Z. Pd kesemt ii kmi meymik uc terim ksih yg sebesr-besry ked Prof. Drs. Setidji, MS ts bimbigy.
Vol. 4. No. 1, 41-45, Aril 2001, ISSN : 1410-8518 DAFTAR PUSTAKA 1. De R. A, Elemet of Abstrct Algebr, Joh Wiley & Sos, USA, 1966. 2. Frleigh, J. B, A First Course i Abstrct Algebr, Addiso Wesley Publishig Comy, USA, 1994. 3. Hugerford, T. W, Grduete Tet i Mthemtics Algebr, Sriger Verlg, New York, 1984. 4. Risighi M. D, Aggrwl R. S, Moder Algebr, S Chd & Comy Ltd, New Delhi, 1980. 45