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## Main Question or Discussion Point

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?

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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?

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matt grime

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

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matt grime

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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.

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.

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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!

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matt grime

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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).

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- #8

matt grime

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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.**

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

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- #9

mathwonk

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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.

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