[ismailmn_1]

(1) suppose R is aring with x square = x for all x belongs to R . Show that R is a commutative ring ?
(2)show that any finite field is of order p to the power n , where p is prime ?
(3) let R be an integral domain and let phi be a nonconstant function from R to the positive integers such that phi(xy) = phi(x) phi(y) for all x,y belongs to R . If x is aunit in R , show that phi(x) = 1?
(4) let R be acommutative ring with 1 and let p an ideal of R . Show that p is a maximal
ideal iff R/P is afield ?
(5) let R be aring and let P be an ideal of R . Show that R/P is commutative iff rs--sr belongs to P for all r and s in R?
(6) find all ring homomorphisms from Z3*Z4 to Z2*Z3 ?
(7) show that a PID is a Noetherian ring ?
( 8 ) let R be aring with 1 which has no zero divisors , and let S be asubring of R with identity e . Show that e=1?
(9) find [ Q(the sixth root of 2 ):Q] ?
(10) find the minimal polynomial of the third root of 2 + the third root of 3 over Q ?
(11) construct afield with 27 elements ?
(12) find the splitting field E of the poly[( x to the power 5) --1] over Q ?

[CaptainRich]

Only twelve questions for today's quiz?
Have you read this:
https://www.askmehelpdesk.com/math-s...-b-u-font.html

[galactus]

I doubt if anyone here feels like doing someone else's group theory homework. Let us see your workings.

But I will tell you this. For #1.

An element x of a commutative ring is idempotent if