Math Is Fun Forum

(1)
Let B:=K[x,y]/(y^2-x^2-x^3) and A = K[x], where K is an algebraically closed field .
Regard B as an extension over A.
For all prime ideals p in A,determine prime ideals of B lying over p.

This problem appears in my text book in exercises after the section " integral extension".
But I don't have a clue how to solve this... .
I am unfamiliar with problems  concerning " quotient rings of polynomial rings".
Hope someone can explain this in some detail.

(2)
Let B be a commutative ring with identity and

be a subring of B(1 in A).
Let q be a prime ideals in B and

.
Then p is an prime ideal in A.
Consider two local rings B_q and A_p.
I want to identify A_p as a subgroup of B_q by a/s -> a/s,where a in A,s in A\p.
But I can't show that this homomorphism is 1 to 1.
Need some help.

(1)Is there exists a polynomial

such that its splitting field

is contained in a radical extension

over

,but

is not radical over

?
(2)suppose

.
    What does "Solve for a by radicals" mean?

Thanks for any help.

Let

(primitive 11th root of unity).
Let

(primitive 5th root of unity).
Determine all intermediaye fields between

and

.

I have shown that actually

,(

(primitive 5th root of unity)),
and

.
So after a little calcualtion,we have

,which is isomorphic to

.
Hence,the subgroups of

are

and

.
Now comes the main problem,how to find the fixed fields of these two subgroups?

Thanks for any help.

Which roots of unity are contained in

,where  is a square free integer?
Thanks for any help.

1.What is the commutator group of A4?
   Is there any simple idea to solve this problem?
2.Determine all conjugacy classes of 4*4 matrices A over R such that A^3 = A
   I have seen the word "conjugacy classes" in group actions.
   But I don't understand what it means here.
3.Show that there exists a subgroup H of a group G such that G is finitely generated but H is not finitely generated.
   I know G must be non-abelian.but I can't find one.

Thanks!

Let G be a solvable group of order n.
Show that there is a sequence of subgroups G = G0 > G1 >...>Gn ={e} such that for all i,
Gi+1 is normal in Gi and Gi/Gi+1 is cyclic of order pi for some prime pi.

Let SO(n,R) := {A∈ GL(n;R)|A*At(transpose) = I}
Let T,D, I be the subgroups of SO(3;R) that preserves regular tetrahedron, octahedron, icosahedron respectively.
Determine their cardinality.