S;::l: Lagrange's theorem in groups: Prove that every group of prime order is cyclic.
by 4 9 & 8 G)ignd -[“ t
o) e |4 i 3 ; 2 )aretwo permutations
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b) If o~
&, 0 AR R,
in Se. Then find s o ?
U o
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3:'*':)) = efine an lntoaral do ?v‘i: m'gwe &
Prove that a commutative ring with unity is a field if and only if it has no non-trivial ideals.
M) Def\ne Rjngi homomorphism. Sh y ly t?Vl.al , .
ey oW,
d(a+ib) = (a, a) for all a, b ? R. Let f: C ? M2(R) given by f(a+ib) = ... Show that f is an Isomorphism of ...
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b) If f(x) ? F[x] and f(x) is of degree 3
and Ol?w it | szm 7 F. '} then prove mm s wreducio\e over ¥
Vfg a) State and .
orem of integral calculus.
b) Define the i) pa tnr' , d interval [a, b] i) n
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orm sums
and iv) Riemann integrals (lower and upper).
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