Old Exam Papers Dec. 2010 - Video

(b) For any real number a,b ? R , we define
a ? b ? a ? b ? 1 and a . b ? a ? b ? ab,
then
show that R, ? , .
b g is a field.
;fn a ? b ? a ? b ? 1 rFkk a . b ? a ? b ? ab gSa
ftlesa a,b ? R okLrfod la[;k,¡ gSa] rks fl) dhft;s
fd bR, ? , . g ,d {ks= gSA
5. Prove that every integral domain can be embedded into a
field.
fl) dhft;s fd fdlh Hkh iw.kkZadh; izkUr dks ,d {ks= esa
vUr%LFkkfir fd;k tk ldrk gSA
6. (a) Prove that every quotient ring of ring R is a
homomorphic image of R and every ideal of a ring R
is kernel of homomorphism.
fl) dhft;s fd fdlh oy; R dk izR;sd foHkkx oy;]
oy; R dh ledkfjd izfrfcEc gksrh gS rFkk R dh
izR;sd xq.ktkoyh ,d oy; lekdkfjrk dh vf"V gksrh
gSA
(b) Prove that a field F is a vector space over any subfield
S of F.
600 4 MT-07
Papers (A) (December) 2010
(357)
(b) Prove that every infinite cyclic group has two and
only two generators.
fl) dhft;s fd izR;sd vifjfer pØh; lewg ds nks vkSj
dsoy nks gh tud gksrs gSaA
3. (a) Prove that every homomorphic image of a cyclic
group in cyclic.
fl) dhft;s fd izR;sd pØh; lewg dk lekdkjh izfrfcEc
Hkh pØh; gksrk gSA
(b) Prove that the order of each subgroup of a finite
group is a divisor of the order of the group.
fl) dhft;s fd ,d ifjfer lewg ds izR;sd
milewg dk xqzikad ml lewg ds xqzikad dk Hkktd
gksrk gSA
4. (a) Prove that the subgroup N of G is a normal subgroup
of G if every left (or right) coset of N in G is a right
(left) coset N in G.
fl) dhft;s fd fdlh lewg G dk dksbZ milewg N
,d fof'k"V milewg gksrk gS ;fn vkSj dsoy ;fn
izR;sd oke ¼;k nf{k.k½ lg&leqPp; nf{k.k ¼oke½
lg&leqPp; gSA
MT-07 3 PTO