Isomorphic Polynomial Rings in F_5[x]

In summary, the conversation is about proving that two rings, F_5[x]/(x^2 + 2) and F_5[x]/(x^2 + 3), are isomorphic. The solution uses the map x \rightarrow 2x to define the isomorphism, but there are some points of confusion. The solution also proves that U(a + bx) = a + 2bx is such an isomorphism, but there is a discrepancy in one of the terms. The solution suggests setting x^2 = -3 = 2 (mod 5) to resolve this issue.
  • #1
catcherintherye
48
0

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

1. What is an isomorphic polynomial ring?

An isomorphic polynomial ring is a mathematical structure consisting of a set of polynomials with coefficients from a given field, along with operations of addition and multiplication. It is isomorphic to the quotient ring of a polynomial ring modulo an ideal.

2. How is an isomorphic polynomial ring different from a regular polynomial ring?

An isomorphic polynomial ring is a specific type of polynomial ring that preserves certain properties of the original ring, such as the structure and relationships between elements. It is essentially a "copy" of the original ring with the same algebraic properties.

3. What is the significance of isomorphic polynomial rings in mathematics?

Isomorphic polynomial rings are important in abstract algebra and algebraic geometry as they allow for the study of polynomial equations and their solutions in a more general and structured way. They also have applications in coding theory and cryptography.

4. How are isomorphic polynomial rings related to isomorphism?

Isomorphic polynomial rings are related to isomorphism in that they are isomorphic to the quotient ring of a polynomial ring modulo an ideal. This means that they share the same structure and algebraic properties as the original ring and can be thought of as equivalent in some sense.

5. Can isomorphic polynomial rings have different degrees?

Yes, isomorphic polynomial rings can have different degrees. The degree of a polynomial ring is determined by the highest degree of the polynomials in the ring, and this can vary depending on the specific polynomials and ideal used to create the isomorphic ring.

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