Isomorphic polynomial rings

[catcherintherye]

I am required to show that F5[x]/(xsqd + 2) and F5[x]/(xsqd +3) are isomorphic, any hints on how to go about this question?

 

[matt grime]

Hint: 3=-2 mod 5. I can give a second hint later if need be

 

[catcherintherye]

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?

 

[matt grime]

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.

 

[catcherintherye]

also I've been given as a hint that 'both dim=2 over F3 basis 1,x H=(aij)

elements of matrix a11=1, a12=b, a21=0, a22=a

that we must have 1=1 and i am to check H(xy)=H(x)H(y) so x=ax + b

but i am still confused!

 

[matt grime]

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).

 

[catcherintherye]

i don't get it I've got order x=ordery=8 and so defined a homomorphism H(x)=y and it seems to give a perfectly good isomorphism, so where do all these hints and stuff come in?

 

[matt grime]

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.

 

[mathwonk]

to show rings are isomorphic, find a map from one to the other,a nd then show it is an isomorphism.

a ring map from a quotient ring to another ring i a ring map from th top tht sends to bottom to zero.

i.e. A map R[X]-->S induces one from R[X]/(f)-->S if it sends X to a root of f . that should do it.