Isomorphic polynomial rings

  • #1

Homework Statement



I am required to prove that [tex] F_5[x]/(x^2 + 2) isomorphic to F_5[x]/(x^2 + 3) [/tex]

now I have the solution in front of me so I more or less know what's going on, however there are some points of confusion......


....the solution states that [tex] x \rightarrow 2x [/tex] will define the desired isomorphism. The next line asserts that

[tex] x^2 \rightarrow (2x)^2 + 2, 4x^2 + 2= -(x^2 -2) = -(x^2 +3) [/tex]

[tex] x^2 = -2=3 [/tex] ............?????:confused:



..what is going on here surely [tex] x^2 \rightarrow 4x^2 [/tex]

since [tex] U(x^2) = U(x)U(x)= 2x2x = 4x^2 [\tex]



anyway this is not my only problem with the solution I have, it then goes on to assert that indeed the two rings are isomorphic and further that,

U(a + bx) = a + 2bx is such an isomorphism


The proof of this says

U((a+bx)(c+dx)) = U(ac + adx bcx +bdx^2)= U(ac + 3bd +(ad + bc)x) = ac +3bd +2(ad+bc)x = *1

U(a +bx)U(c+dx) = (a +2bx)(c+2dx) = ac + 2adx + 2bcx + 4bdx^2
= ac+ 3bd + 2(ad + bc)x = *1

fantastic! execept look at term 4 2 lines up, 4bdx^2 = 12bd=2bdmod 5 and not =3mod5.....:confused:

so what am i missing here?
 
Last edited:

Answers and Replies

  • #2
5
0
I don't quite understand your first problem, but for your second problem you should set [tex]x^2=-3=2(mod 5)[/tex].
 

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