In summary, the person is asking for a hint on how to show that F5[x]/(xsqd+2) and F5[x]/(xsqd+3) are isomorphic. They are given two hints, one being that 3=-2 mod 5 and the other being that 4=-1 mod 5. The hints are meant to help the person get started, and they should be able to figure out the rest on their own.f
okay, so i have done similar questions where i show e.g F5[x]/xsqd+3x+3 isomorphic to C24 would it be an appropriate course of solution to attempt to show that both of the above fields were isomorphic to a group and then conclude isomorphic to each other?
No. You're asked to show they are isomorphic as polynomial rings. That means more than just isomorphic as groups (under what operation?). So your previous example is not correct, unless you want to cite some big result to do with fields or something.
You can do this by just writing down the isomorphism. Hint, again: 3=-2 mod 5.
EG. Why is Z[x]/x^2 isomorphic as a ring to Z[x]/(x-2)^2? Just send x to x-2.
What's confusing? You're writing it as a vector space, and writing down a map. You're asked to show it is a ring homomorphism too. It wouldn't be the way I'd do it (though it is equivalent).
If you know these are fields, it suffices to count elements. If you don't then you need to write down a ring homomorphism, so you need to get your hands dirty and do it. Any ring map sends 1 to 1, and must send x to ax+b for some choice of a and b. So figure out what, if any, choices of a and b mean this is an isomorphism.
There are only two things to remember: 2=-3 mod 5 (why do I keep writing that I wonder...) and 4=-1 mod 5 (that is the second hint).
What is x, what is y? So what that the order of x and y are the same? What do you mean by order? I will guess that you mean the map from
F_5[x]/(x^2+2) ---> F_5[y]/(y^2+3)
sending x to y is an isomorphism. This I doubt. For instance this map sends x^2+3 to 0, and x^2+3 is not zero in F_5[x]/(x^2+2). It is in fact equal to 1, so your map sens 1 to 0.
Let me explain the hint. The two rings are isomorphic to F_5[a] and F_5 where a^2=3=-2 and b^2=2. Notice that -1 is square mod 5 so multipliying a by a square root of -1 is an isomorphism (it is an invertible ring homomorphism). This corresponds to multiplying x by the square root of -1 in the polynomial ring case.