THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/2008, pp

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1 THE PUBLISHIN HOUSE PROCEEDINS OF THE ROMANIAN ACADEMY, Seres A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/8, THE UNITS IN Stela Corelu ANDRONESCU Uversty of Pteşt, Deartmet of Mathematcs, Târgu Vale Street, No, Pteşt, Argeş, ROMÂNIA E-mal: Let be a rme umber The rg has bee efe [6] as beg the comleto of the fel of algebrac umbers wth resect to the so calle sectral orm Sce ths rg seems to be terestg from may ots of vew, we wat to cotue to vestgate ts roertes of t The am of ths aer s to vestgate the uts a the regular olyomals of Key wors: seuovaluato, sectral etesos, "Krull toology", sectral orm, regular olyomal INTRODUCTION I [6] t s efe the rg Ths aer s a atural cotuato of our revous aers [], [6] We te here to escrbe the uts sets a the regular olyomals of The aer s ve to four sectos The seco secto resets the otatos, eftos a bass re-sults I Secto 3, the uts are stue Fally, Secto 4 we stuy the regular olyomals a let Deote by v etes v NOTATIONS, DEFINITIONS AND BASIC RESULTS the fel of ratoal umbers a by eote the valuato o a fe closure of t Let be a rme umber efe by t [] Also eote by v a fe valuato o whch Deote by al ( / the alos grou of automorhsm of For ay, eote by v the valuato o efe by ( v( v( ( for all The v s also a eteso of v to a ay valuato Let v o whch etes v s of ths form be the fel of -ac umbers a let us cotue to eote by the uque valuato o whch etes uque valuato o Fally, let C v Also let whch etes v be the comleto of uque valuato o C whch etes The value fel be a fe algebrac closure of, a cotue to eote by v the wth resect to v a aga cotue to eote by v the v (, v v C s calle the comle -ac fel

2 Coser the value fel (, v valuato v o Stela Corelu ANDRONESCU Deote by w the sectral seuovaluato o uce by the Accorg to [6] for ay oe has f ( w ( v ( f ( v( We shall say that w s the sectral eteso of v to Deote by the comleto of wth resect to sectral seuovaluato w above efe Also, eote by w the uque etesos of w to Sce w s ot a valuato, the s ot a tegral oma ([6] Corollary 5 Deote by the sectral orm o whch corresos to w Accorg to [6], for ay oe has Let For ay, we efe the ma θ C the followg way w ( : Let { }, where { }, s a Cauchy sequece the class of, for ay,, The { ( } s a Cauchy sequece ( C, v Let us ut θ ( s class there Shortly we wrte θ ( The elemet s well efe by ts comoets { } I fact we have su( If, the lm, where { } w w( f ( τ v( τ If,, B Ba (, r { a < r} a (, ( v (, where s the absolute value of C s a Cauchy sequece of wth resect to w Ths meas that a a a f r > s a real umber, we mark wth the oe ball wth the ceter a a the raus r, a wth B B ( a, a r the close ball wth the ceter a a the raus r r { } Deote wth B( a, ε { a < } (, ε { } C r the oe ball wth the ceter a a the raus r, a B a C a r the close ball wth the ceter a a the raus r Let us coser, ( a ρ > a real umber If for ay we have < ρ, the < ρ ([3] Lemma The set C { } Let roertes: y z ε ; ( s comact ([] Remarks 39 y a ε > be a real umber The there ests a elemet z wth the followg eg ( z eg ( z eg ( ([] Proosto 4 y 3 THE UNITS IN Deote by U the grou of uts of, that s U ( y sothat y { } We have the followg theorem of characterzato of the versbles elemets of the rg

3 3 The ut Theorem 3 Let us coser ( from The elemet s versble f a oly f, for ay Proof Let the elemet be versble The there s y ( y, y so that y Therefore ( ( y ( y, that s y, for ay, whch shows that for ay Coversely, coser, ests ( Coser the set C ( { }, so that, for ay We shall rove that there from C Usg Remarks 39 of [] we have that the set C ( s comact a close, a the zero elemet oes ot belog to C ( We state that there ests a real umber ε >, so that B(, ε C ( Iee, f for ay ε > there s a elemet a C ( B(, ε, the we fer that there ests a sequece { a } C ( coverget to zero ε But ths case the zero elemet s set C (, whch s a cotracto So that let us coser ε a real umber wth B(, ε C( The for ay we have ε a so Let { } be a sequece of elemets ε wth lm We kow that for large eough we have, so we have ( ( su( su su a so Coser δ > be a small real umber The for large eough we have < δ, or ( ( < δ, we have ( < We ca coser < So we obtae (, for ay δ Let us coser the sequece { y } For δ ε, a the we have ( ( ε, for ay a large eough +, where ε, where y We shall rove that ths sequece has a lmt large eough, sce ( ε we have ( y ( y + + ( + ( + ( ( ( ( ε su ( su + ε Hece, the sequece Let y lm The we have + s coverget the sectral orm y lm lm, that s y, y

4 Stela Corelu ANDRONESCU 4 I orer to reer evet both the toologcal a the algebrac asects of the roblem of uts, we shall gve two roofs for the followg theorem Theorem 3 If a <, the the elemet s versble Proof The frst roof of ths theorem s base otheorem 3 Iee, f < the su( < a so <, for ay But the, for ay whch roves that, for ay a so the elemet s versble The seco roof s classcal a s base o the fact that, f s coverget Let A be the sum of ths seres A lm lm <, the the seres We have (, but the A lm ( ( + ( ( a so A The roblem ow s to f out how may uts there are Deote by θ ( where C C \{} For ay Theorem 33 The fucto θ : U C efe by θ ( s surjectve Proof Let z C a We shall rove that there s a elemet U so that z For ths we shall use Proosto 4 of [] So coser the sequece { }, we choose a elemet eg eg z eg ( ( ( We state that the sequece { }, accorg to Proosto 4 of [] so that coverges to z Iee z + z z + z ma,, z+ z + Sce lm( z + z t s ferre that ma ( + z+, z+ z, z ( lm + θ : U C, z so that lmz z a so the sequece { } z < a s coverget the -ac orm But, accorg to Proosto 4 of [] t ca be ferre that the sequece { } also coverget the sectral orm, sce v( + v( ( + (, for ay [6] Hece lm Remark 34 Let su f z, for ay a thus we have rove that the fucto θ s surjectve U We have su ( a sce ( Sce the sets C ( { } a C ( { } ferre that su( C ( as well as that C (, we obta s are close a comact, t s mmeately f, where C ( { }

5 5 The ut Now let su a f We state that Iee, we suose that < ( ( But we have <, because But or Thus we have obtae a cotracto a ths shows that the equalty ( s ot true So, that s 3 Deote by Ia ( { a U }, or Now we te to eame the ots of cotuty for the fucto f : I( a f( a, where a We have a U f a oly f a, for ay So a >, for ay Proosto 35 If Ia (, the ( > Proof We suose that f ( a f a Sce s a comact sace the "Krull toology", ay sequece has a lmt ot Coser the sequece { } so that ( a lm efe by Sce { } ( C a { a } C ( C ( C ( a from the sequeces { } a { a } we ca etract some coverget subsequeces { } { } { a } { a } We suose that lm a lm a a a Let, so that lm (3 Sce ( a t a, a the set a are close sets, we fer that t ( a lm, a takg to accout the relatos ( a (3, we fer that, a a, that s a Thus we have obtae a cotracto Therefore, f Ia (, the f ( a > Proosto 36 The set I( a s a oe set Proof Let Ia (, that s f ( a > Deote by f ( a Let y B,, that s y < Sce y su y a y < we have y <, for ay For y I( a, that s f ( y a > we obta y y a + a Thus ( > f y, that s the set Ia ( s a oe set

6 Stela Corelu ANDRONESCU 6 f Remark 37 Takg to accout all we have rove above, the set of ots of cotuty for : I( a, f( a {, where a, s the set f ( a > } 4 REULAR POLYNOMIALS IN Let us coser [ X ] The elemets of [ X ] are olyomals of the tye f ( X a + ax + + a X, wth coeffcets Coser f ( X a + ax + + a X The f ( X( a + ( a X + + ( a X are calle the rojectos of f o C, for ay For f, g [ X oe obtas ( f g f g ] Defto 4 f [ X ] s calle regular olyomal f there s a ozero vsor Proosto 4 The olyomal f [ X ] s regular f a oly f f for ay Proof Let us coser the regular olyomal f [ X ] efe by f ( X a + ax + + a X If for, we have f, the f ( ( X a + ( a X + + ( a X, we euce that ( a ( a ( a Accorgly [6], there ests a elemet e so that e e a e a, for,,, Let g e X We have g f ( e X f ea X + ea X + + ea X +, whch s a cotracto Hece f, for ay Coversely, let f for ay We suose that f g, for f, g [ X Oe obtas: f g ( f g, for ay Sce f, we have g, we euce that g Hece the olyomal f [ X ] s regular Corollary 43 Let us coser f ( X a + ax + + a X, wth a, for,,, If ests a elemet a U(, the f s regular Proof If f ( X a + ax + + a X a a U (, the ( a, for ay, we euce that f( X, hece the olyomal f s regular Remark 44 I artculary, the moc olyomal f [ X ] s regular ] REFERENCES AMICE Y, Les ombres -aques Presse Uv e Frace, Collecto Su, 975 ANDRONESCU SC, Sets a elemets Rev Roumae Math Pures Al, 49, 4 (4, ANDRONESCU SC, O coverget seres Mathematcal Reorts, 5 (55, (3, -7 4 ALEXANDRU V, POPESCU N, ZAHARESCU A, O the close subfels of C J Number Theory, 68, (998, POPESCU EL, v -bass a fuametal bass of local fels Rev Roumae Math Pures Al, 45, 4 (, POPESCU EL, POPESCU N, VRACIU C, Comleto of the sectral eteso of -ac valuato Rev Roumae Math Pures Al, 46, 6 (, Receve Setember 9, 8

ELEMENTS OF NUMBER THEORY. In the following we will use mainly integers and positive integers. - the set of integers - the set of positive integers

ELEMENTS OF NUMBER THEORY. In the following we will use mainly integers and positive integers. - the set of integers - the set of positive integers ELEMENTS OF NUMBER THEORY I the followg we wll use aly tegers a ostve tegers Ζ = { ± ± ± K} - the set of tegers Ν = { K} - the set of ostve tegers Oeratos o tegers: Ato Each two tegers (ostve tegers) ay