Prove that and
.
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मान लीजिए कि . जाँच कीजिए कि D एक UFD है या नहीं।
heck whether is a subring of the ring
or not. If it is, check whether or not it is an ideal of the ring also. If I is not a subring of the ring, then provide a subring of the ring.
Show that there are infinitely many values of for which
is irreducible in
.
Check whether (X, ) is a group or not.
Let be a finite abelian group and
. Prove that
.
Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so.) (20)
i) , where
is the Euler-phi function.
ii) If G1 and G2 are groups, and is a group homomorphism, then
.
iii) If G is an abelian group, then G is cyclic.
iv) If G is a group and , then
.
v) Every element of Sn has order at most n.
vi) If R is a ring and I is an ideal of R, then and
.
vii) If is a product of an even number of disjoint cycles, then
.
viii) If a ring has a unit, then it has only one unit.
ix) The characteristic of a finite field is zero.
x) The set of discontinuous functions from [0, 1] to form a ring with respect to point-wise addition and multiplication.
) Define a relation R on , by
.
Check whether R is an equivalence relation or not. If it is, find all the distinct equivalence classes. If R is not an equivalence relation, define an equivalence relation on .