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/sci/ - Science & Math


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>> No.15151776 [View]
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Full answer:
>GF(25) and GF(5)[x]/(x^2-2) are not isomorphic, which means that they are not structurally identical.

>GF(25) is a finite field with 25 elements, which consists of the integers from 0 to 24, with the operations of addition and multiplication defined modulo 25. It is a field of prime order and it is used in fields such as error-correction codes and cryptography.

>GF(5)[x]/(x^2-2) is a finite extension field, also known as a Galois field, which is a polynomial ring over the field GF(5) modulo the polynomial x^2-2. It consists of all the polynomials in x with coefficients in GF(5) that can be written in the form a + bx, where a and b are in GF(5), and when multiplied by x^2-2, it equals 0. The operation in this field is polynomial addition, polynomial multiplication and polynomial scalar multiplication.

>The two fields are not structurally identical because they have different numbers of elements and different operations. GF(25) has 25 elements and two operations, while GF(5)[x]/(x^2-2) has 25 elements, but it has three operations: polynomial addition, polynomial multiplication and polynomial scalar multiplication. Additionally, GF(25) has a prime number of elements and it doesn't have any non-zero divisors, while the elements of GF(5)[x]/(x^2-2) are polynomials and they do have non-zero divisors.

>Isomorphism is a property that relates two algebraic structures that are identical in structure, and it is not the case for these two fields.

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