Summary

These lecture notes from R&F (possibly a course in Rings and Fields) cover fundamental concepts in abstract algebra, focusing on rings, fields, and subfields. Mathematical proofs and examples are presented.

Full Transcript

Rn n - 6 6 , 1 1 % 1 1 i f : 1 ¥ I i I s b n # j £ g H " -...

Rn n - 6 6 , 1 1 % 1 1 i f : 1 ¥ I i I s b n # j £ g H " - 9 - ⇐ i s # I §i,-no b i g # ↳J i a j i i s ⇐ (p→q=-q→-p) [email protected] v i s i t I n @ I 0 2 a. b e n § i n a b i. a -b- = abt = P = T I s ⇐ I s a y # 6 - To set A d a ' s ↳ 'l d ' s , Tibet. @ I s I T. b g I s - 1 - - 1 I n ⇐ £ 74.d e # s i p i - N g i s s n o i g I ⑦ I s T i m b s 1 6 1. I I I i s I n a i d. I f - I d @ ¥ # I i 0.61, o < a e n A E K ⇐ o f I E En a i I E j j yeK - a = ' axtny x. ⇐ god Cn,a t l : ' t.5-a.at-0 s a t i n a t * I = ascent = T t a - I - I i. & I n i. I f i f ¥125 7 £ @ i s c A 3 % § Ep P I s i s i n s is A % r i c h e s i s 5 1 k ¥ i. ↳£ 1 I T i f R:{rot, r , ,... , rn..} I l I I I I. A.b ag I t s R o ! I I ! WE ftp.gjssoio.w.si R § Of A E R § a i ¥ , I i. a s a R i t a , a r, s.... arm,} i Yippee µ I t R t h d R I k % ; ← (Rs.) 5 ¥ I s a R s R i d ' ↳ a o ⇐ Rent A area E R [email protected] § CARE R g it.iq R ie lo...... n i b s. I A. i f IER O i ↳ a wofri ok Isis, a r i = l & o t.fm#- @soaox'@ F D D b o o d I @ I s I b i d § ' ¥ : I I s t o p £8151 died-Isis- c h a r CR) = O o r c h a r (R) = p I % i f R a. I £ 1 - : - # char (R) = n I i t.si, i t ' r n # O I I , I i s e e n i I. i f s e e n ↳ i f D = n, n , i l c h.,n , < n (n,-1) ( n i l ) = (nine)-l = n. I = O → ① a i d g-ga# I s in" W i a n d s e l f t. b r t o ? ¥98.006,59 n 0£51 H o g lol "n-1=0 is#& E I ① I > b i s i ← ( n ,. 1) (nail) = O I d g ' s A §'S, D R i s , a t I r i s ' s (n.. 1 7 = 0 si (n,-17=0 ⇐ I I I I. i s , n,,n, < n & [email protected] n. 1 = 0 i I i i i s , @ signs n @s i g n , I 0 £ , o l a t i. ↳ I @ I t , j § I s j s p 6 £ A - R - I " I I → s ? I d. - t o R £ 1 : 4 ¥ ; i o s > I n- 1 1 6 % 1 , ↳ is i s m ' s ' d gis t ( R , t..) t b h ① A # O ab= A C £ 5 1 5 1 ② , t o o ↳b l i s s o f g o. I M. i s R w ; D i o s , i d f ' d s.gs/s-rIR ↳ i s ; D @ ← ① also r absac , a b - a c e o ⇐ a # O r a C b - 0 = 0 ← b-C e o ⇐ b = c ⇐ Digit@boi ↳ A ①← ② R § ' t F a n i 1 4 I I n o s b i o , BER A. £ ⇐ a b = 0 a. 0 = 0 I & I a b = a. O ⇐ £4,18, b - o ⇐ btog.gl. ibkasjwzssofs.gl', R § i s # b s.-24 R i - 1 6 1 1 l R , - 1 ,. i s ' d i. J I Is d i d so bt i l l - a t R j - A (Ita) ↳ & C E O J u d (Italll-a) = l - a + a-a' = L (Lalata) = I + a-a-a-=L t a x t b - 0 @ s I p t. s.. I T i t § Adds (Foti) i s § a t O t b. § l Rit,-' t h ' § / CF, + ,. ) i s t.w s : & ' a e F-{03 a" E F A s axtb=0 a x = - b b i a l y , lat-b x = ail-b) j g i s - I. T L. A AER-903 ⇐ R i g g ' 'ER. - s k y 1- t o , a- - R i b s subfields i.sk?dJ--S1 ↳ J I. £ (F,+ ,. ) 0 6 1 1 i § Wi F ↳ E 4 6 ¥ - s h a d , ↳ T H F g (subfield) p j s. i f i f ( E. +,.) - # E E F i F A F G H i 1 ¥ ⇐ H I J A F ↳ § o f E 0,6 a. BEE A a - BEE o -BIO, a. B E E ' s ab-'EE o O D thy-§, ¥. A t. g r @ I t TK ↳ v i. A Q : @ I - ¢ ↳ B i g R O ' IR ↳ "b. 'Qerz) a i I i " " T /QCF)={atbrzia.be/Q} o ' s ¢ I QCE) E R... H i [email protected] O c r ) wing..-just c- X = a t b - , y = c + ditz Ky-Catbird-Cctv) = la-c) + Cb-d) E E Q c fc ) Ky's Catbr). (c + dry" = catbird. 1¥,.E¥¥ ) = Catbird ( E t ) (a+ bit). = ( ¥ , -¥12 t ) c- Ou E Q =(¥¥-ZEE)t(¥zd.-1¥) FL N E X.IR@IRA O c k ) i. I I I. % j s @ b § I £ J l 1 µ g (prime Field) [email protected]'t i f t.HR W i &? J A Q , R I D d i s i d K i d @..?%i ' A - I Q i t £ 1 1 o f @ Dio ÷§÷. i n d w ¥ , [email protected] i d - j. C 714, I 716,... r£←g3y%i%←3s←£ "b←%oµibI:b i s I ←. a → → (And,&,bTtYoi i n" To i dei)

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