Isomorphic Polynomial Rings in F_5[x]

[catcherintherye]

Homework Statement

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

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 $$x \rightarrow 2x$$ will define the desired isomorphism. The next line asserts that

$$x^2 \rightarrow (2x)^2 + 2, 4x^2 + 2= -(x^2 -2) = -(x^2 +3)$$

$$x^2 = -2=3$$ ...?



..what is going on here surely $$x^2 \rightarrow 4x^2$$

since $$U(x^2) = U(x)U(x)= 2x2x = 4x^2 [\tex]<br /> <br /> <br /> <br /> anyway this is not my only problem with the solution I have, it then goes on to assert that indeed the two rings are isomorphic and further that,<br /> <br /> U(a + bx) = a + 2bx is such an isomorphism<br /> <br /> <br /> The proof of this says<br /> <br /> U((a+bx)(c+dx)) = U(ac + adx bcx +bdx^2)= U(ac + 3bd +(ad + bc)x) = ac +3bd +2(ad+bc)x = *1<br /> <br /> U(a +bx)U(c+dx) = (a +2bx)(c+2dx) = ac + 2adx + 2bcx + 4bdx^2<br /> = ac+ 3bd + 2(ad + bc)x = *1<br /> <br /> fantastic! execept look at term 4 2 lines up, 4bdx^2 = 12bd=2bdmod 5 and not =3mod5...<img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f615.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":confused:" title="Confused :confused:" data-smilie="5"data-shortname=":confused:" /> <br /> <br /> so what am i missing here?$$

 

[Bavon]

I don't quite understand your first problem, but for your second problem you should set $$x^2=-3=2(mod 5)$$.