R- m mn t -R, as usul

Tua is a mo n s R ln)=a ieu as nd R

+ no n

Sum 1n R

Po ssicies Brn ( r) suu -n -(1+ +

(a) p(7¢) >R, hu p =

R R orduns7 as sd tranas2, 2L% J, a, R, C, Z().

(6) 6) is o i.cse. ku (e)#o. Sia Z isa PLO, a

s s prinde , L (e)= (n) ,no ()> (-n)

Ten

Tu inas 2 in R sa Aie esius moulo n

o, n- =0 inR e : R>72/(a),

7L/n ra um a A, as a

R Coadaas e tdhn Z

su u dauisans

A SSu R i AI K= F

P aoe nekm 0 P°n

siu QF no F 2Z ,

en F as

Fe- P Fp- o, P RA ueidey m P a

Exe F A FL polunomi s tn F, ac (F Cz)) -iSnahard F Rac (F(z1) ads ,), () e (x)

Coe de As in F.

tequvaara wladan, (e) f o () #o

-2 Prop 1 1dgra doain R is a sus RAF, tluu

aO mo monpm ac (R) F L xtnds

inus n R F

rac R SudhB's u 1

B? Ele menks k oc (R) a pa1s (, ) , f,

abeR B(ab)=Jl) J(L)

xei: )B Sa o momep sn,

2 s R

an

Prope sidos Sans wk ar1ncson

axdsto an neusu. u h Saehons Rrac (R)= 8(R) F.

Q= rac (Z) tx e F eE cF

TF F åt ains suldl 2

Pi RASs.

FF, sa ta clanacktusic isP ch (F=P

h C)Fo a dhannchiis be of F is O

c F, so Rer

xezu ) ?lr loma R is a s a kll , Re

T Smeu dR Al dains R is isomophie &(R)>Yae (R

b)o cld Anks : eT inakU t.

-3 F-sd, Flx]

Savisan Jlad, Gne polyoil s (a)q(x)e Fl*]

und tm oynomi als

&(oc) ()am

Tonsiued q (e),s(x) e dvid (x) (x) m a uwain le

Snsuin on

cz) < g(*) (n ca) don c) ( x)* {(x) (e)=lx)

nm &( a**o,-,"*. a o

9 ( 6."*. *

) = a, ba(b.m x"6z '4 +b). =a,**o, b C +

Can inet in F, mtos tta Psa iA, 6en o i cln tnws

-m

, ta, -*" T. «a-(a,c * +a, b5

- a, 6. b., ) 2

)-a8**)¢ (z) as dau n Proco S adlucto

,z) as a

()q, (()+, (*) )=q (x)a*)ar, (x),12) (2) +, (*) qx)q(x) +2(x) (-2)a(e) = , («) -, (*)

wless 2, LtS

Diso pono i d's -1

a ema

o,,2} 2+2 |

8(e)=2x+x2x +* () +22 -1

2= - mos

ensen 2ct

+0x +2x+x S - -

2 + 2xt x

2 -2x- 2* =mals)

2t

2x*x = (*2x) (xx -t) 2oc+

nenie (L ePhiat is 1)Can di g(x) e)u wen

oun RC«, R a is monde, rud dop coefhao

htnu thl in R

7l). Can t ila 3(x) 2 **/?

is rod Canr t 2t

or uaon , uset do a kld F and polroials tn Flx

Tm F) is prndi pl idorl doan PTb), a k T P Ta an la Te PC*] T=(o),* is prnpal T=(o)

T (o), clasvt a poly n m )eL 0u ma re.

m() CL SSu Pr t P6e) eT\(la)

d senna m (e)

a ma in lL

= q (z)m (x)+ (x) Ja r(x)é da m(z) 9 (x)

(a)eC si e >f(x)-q(* )m (oe )

m (x). (art In)

Colla In F(3) has bim (n() lo), a

= *ta,, x^** amone daA.

m( m (x) ioaals (m(T)), (z) au dshine

Divisb seR S dvdss s cls 3'e

seCr) -Priniga ad attd .

clo re, nLe $Lu)

u H clu Ofr (

D

ta,g ispoyoniol Ac). al, alg C) el, clg - cl -(,3) 3) Smo

San at , d Ia dve Pra

8,8 cds

A'8'1&,Fl)is&n d,d' drktt a a un

See saa ho F-3 P

exists

P

Ju s penpal (F6) is a PId) , T- ( )

(R),lass -()- &F()o V Can dao

cl,ey > cc',ecc" X=alrba acc. bc c'= clactbc')

maEuu) FheAeS, p(x)e F) oa pret -7

no ainds of ma n dscre

T) I2). .. q (a)

n22

Sln P) () - I~

1 a (ne P«}+ G(x) q(«) S a.

al) fc)k (a b(x)g (e) k(x )

ahr egk,la, k halht be = (a kt be)f

en

P ()

To s8 d2moe geneca

Su icuals fa, g ) = (?())+ (g(*o)

Evec T,S 1Is In R I+Tiirjl iel,jeJa dls ToS

tn R

Pncpa Sum a{

(),y) 6), gPM) (R-goe) (A() T

aX ra. = a,\c+ a,-0, z* tQ,0

/o.eF . c

sca oc prena ds Ponaal s

0 opude qc&8,3) , kd dauis To

A&2s 3 , 3)

is ph e man ln now

(3 2 + ) )

Exe (P,g) ad (,)ha

u sa

remasne

(,3) - nea+t

( o neAnailn

ramp F=7/^= 5 -f (2e) 2e 2 x2z+, 1(«)z-**l. Rd (¢a),z(z?))

emann en

(xt2A 2x tl, x'-x+1)

-z +l, T* -+x 3=0 3+X*

cdf(x)g () = *+l

onc V

-2x

2) F F lo, 1) f(x)= *'t*°, (z)= 24X+|

emaan (z, x'2)

(z, z*)

(z ) T

&,au uladres pri

2C

- 10

Cplor vs M u mandn)

F3 Ten Le sist udea polns omials 9r eFLx), m r o Su at o oan

d O -

+

ola osef+ (f) 2ve

has a w etndese

Yemahd

Pco + Wck -q e (4)

3e ) CeQ (E) o set

Such C Vwa

-2e(t) , ht sdf +le)= +e)

Coyess () : uqeseed O a «ll (o miols { Jkra n

(-b2x+ b,x«b. 6,#0 E- 2

6us F:>/e)

4uaduic atx t+(f\

A padns (a,a,)a¬F o,1

telx a or all Rnea 6mh don

&n 3 ic

Cos F

sis 2,z Cocs adad ,4) a;e F

-it n

tus ? 4+4,r

elAs s asre sìdus o ulo P

a ossibu F/( )

EKA F F=Q (= *+x*|.

daaks as paromids o{ Ja at mogt

at,

o ul n K, mulkpas polno mus ,u sals a

2

rvi fon

(24)( -3z) = 2-S -3x -2-Sx -3 (-r-)-2-5�x) =

+S

o 2 +*t(f)

-3t 1f(x\)

(24x)C-3)=4S 1n x+t). Xx = x-- x -l I R/r -

-3-S*t2

3*3* t 3 2) F F to,} o-2z+S

=z+**l As.

R: 7 Ix*xri) c(x)= xt* = -I = (wd 2)

T s a kIS uMBr naAs

IFCxl/x*zed

-12 Rmean yo ro c-a

remain lA, a

x -a) C

eaF(x) F

(x )+c (a-a) gla)+c =

=0gla) +C =C &la) =C

lx)- (z-a)g (*) * Pla) isla)

(z) is avi des (x-a) ,0 remaln is la)

e F Consfe pono is

FLax3-e oses ane

he)t(x -a) k(a) jkn, n, (spo sweh

t(aa

R/z-) S sa IR

2 2

3

Com Son -13

FLx a

Bo au PTDs

Invti Fl-)-F

ae

Sa Conste

(n) (-a) posi unn

ie Pornmial

n m k a)=q(z )h(x)

P° e ite u cGe polynd

23,,7,1,1) P) at.

eFa no e duiGle lnamdy

(uille's la aeks ) (i&L elan.s)

WM C

z1- p.(x).. P«() PiPa i radobn

o C Ju po s

=(-) p P

T =a a P P(z)

T moade esu,

ced ed,q) ecm ,) ec ,3)

opi , an+bm Som a, So a(« ),

alx){r\r b(x)g)