1. ## field ideal question

Let I denote the field generated by x^4+x^3+x^2+x+1 in Z2[x] and F=Z2[x]/I then
.1)F is an infinite field
2)F is finite field with 4 elements
3)F is finite field with 8 elements
4)F is finite field with 16 elements

next question
let G be k group with the generators a and b given by
G= <a,b:a^4=b^2=1,b2=(a^-1)b>
if Z(G) denote the center of G then G/Z(G) is isomorphic to
1)the trivial group
2)C2,the cyclic group of order 2
3)C2 x C2
4)C4

in gate entrance exam

2. Originally Posted by reflection_009
Let I denote the field generated by x^4+x^3+x^2+x+1 in Z2[x] and F=Z2[x]/I then
.1)F is an infinite field
2)F is finite field with 4 elements
3)F is finite field with 8 elements
4)F is finite field with 16 elements
Any element in $\mathbb{Z}[x]/I$ has form $a + bx + cx^2+dx^3 + (I)$. Where $a,b,c,d \in \mathbb{Z}_2$. Furthermore different representations gives different elements. Thus, there are $2\cdot 2 \cdot 2 \cdot 2 = 16$ such elements.

G= <a,b:a^4=b^2=1,b2=(a^-1)b>
Does you really mean $\left< a,b, : a^4 = b^2 = 1, b^2a = a^{-1}b \right>$?

3. Yes you are right.Yes i mean that. Please thanks alot for your help