Isomorphic rings are rings that have the same algebraic structure, meaning they have the same operations and properties. Isomorphic rings are essentially "equivalent" to each other.
The ring \mathbb{F}_5[x]/(x^2+2) is a quotient ring of the polynomial ring \mathbb{F}_5[x]. It consists of all polynomials with coefficients in the field \mathbb{F}_5, where the ideal (x^2+2) is used to define the addition and multiplication operations.
The ring \mathbb{F}_5[x]/(x^2+3) is also a quotient ring of the polynomial ring \mathbb{F}_5[x]. It is similar to \mathbb{F}_5[x]/(x^2+2), except that the ideal (x^2+3) is used to define the operations.
Yes, these two rings are isomorphic. This can be shown by finding an isomorphism (a bijective ring homomorphism) between them.
Isomorphic rings can be used to study different structures in mathematics. By showing that two seemingly different rings are isomorphic, we can apply results and properties from one ring to the other. This can help simplify calculations and proofs in various areas of mathematics.