Abstract Algebra Answers

Let R= Z[root 2] and M={a+b root2€ R|5|a and 5|b}.
i) Show that M is an ideal of R
ii) Show that if 5 |a or 5|b , then 5|(a2 + b2), for a,b € Z
iii) Hence show that if N is an ideal of R properly containing M, then N= R
iv) Show that R/M is a field , and give two distinct non- zero elements of this field

Let D={f(x,y)+g(x,y) i |f , g€Z[x,y]}subset to C[x,y]. Check whether D is a UFD or not

Define a relation R on Z by R={(n,n+3k)| k€Z}.
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 Z.

a) Check whether or not A={z€C*||z|€Q} is a subgroup of
i) (C*, .)
ii) (C ,+)

b) Let (G, .) be a finite abelian group and m€N. Prove that
S={g€G |(o(g),m)=1}<_G

Explicitly give the elements and structure of the group Sn/An,n>_5

Find a group G and homomorphism phi of G ......

find all the units of Z12

Explicitly give the elements and structure of the group Sn/An, n≥5.

1. 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) f(n) = n −1 "nÎN, where f is the Euler-phi function.
ii) If 1 G and 2 G are groups, and 1 2 f :G ®G is a group homomorphism, then
o(G ) o(G ) 1 2 = .
iii) If G is an abelian group, then G is cyclic.
iv) If G is a group and HDG , then | G: H | = 2 .
v) Every element of n S has order at most n .
vi) If R is a ring and I is an ideal of R , then xr = rx " xÎI and rÎR .
vii) If S (n 3) n sÎ ³ is a product of an even number of disjoint cycles, then sign (s) =1.
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 R form a ring with respect to pointwise
addition and multiplication.