1)consider Z24 as additive group modulo 24.Then find the number of element of order 8 in group 24 ?
2)Consider Z5 and Z20 as ring modulo 5 and 20.Them the number of homomorphisms
p:Z5-->Z20 is?
3)if Q is a field of rational numbers and Z2 as a field of modulo 2 let
f(x)=x^3-9x^2+9x+3 then witch of following is correct.1)f(x) is irreducible over Z2
2)f(x) is irreducible over both Q and Z2
3)f(x) is reducible over Q but irreducible and Z2
4)f(x) is reducible over both Z2 and Q
please explain with detail.Thanks in advance
(these questions asked by indian institue of technology delhi asked in gate test )
Since f is of degree one, f is reducible if and only if f has a linear factor.
Now recall that x-c is a linear factor if f(c)=0. So basically we only need to check if there is an element c in the field that make f(c)=0
for Z2, we have f(0),f(1) not equal zeero so f is irreducible there. For Q, by Gauss Lemma we need only to check if f has an integer root. This integer root must divides the constant term, i.e. 3. So you need to check f(1),f(-1),f(3),f(-3). If one of them is zero then it is reducible but if not then it is irreducible.
Let . This is a subgroup of order . Any element of order 8 must lie in this subgroup. Since is a generator it means all other generators i.e. elements of order 8 are where . Therefore the are such elements.
Hint: If is a homomorphism then it is completely determined by its value on . Find the possibilities that this can be.2)Consider Z5 and Z20 as ring modulo 5 and 20.Them the number of homomorphisms
p:Z5-->Z20 is?
is irreducible over if and only if it has no zeros in . Likewise for .3)if Q is a field of rational numbers and Z2 as a field of modulo 2 let
f(x)=x^3-9x^2+9x+3 then witch of following is correct.1)f(x) is irreducible over Z2
2)f(x) is irreducible over both Q and Z2
3)f(x) is reducible over Q but irreducible and Z2
4)f(x) is reducible over both Z2 and Q