[Abstract Algebra] Maximal Ideal

[cummings12332]

Homework Statement

Let F be the field and f(x)=x-1,g(x)=x^2-1 and F[x]/(f(x)) is isomorphism to F, is it g(x) maximal??


2. The attempt at a solution

I will say no.Since g(x) is not 0, the dieal (x^2-1) in a prime idea domain F is maximal iff (x^2-1) is irreducible.
And we say (x^2-1) is irreducible if it is not a unit, but x^2-1=(x+1)(x-1) implies that either (x+1) or (x-1) is a unit.
but I can find a taylor expansion of 1/(x^2-1) which means (x^2-1) is a unit, contradicts irreducible

is my idea right ?

 

[mighty2000]

it is important to accurately evaluate and respond to forum posts. In this case, I would like to offer some clarification and additional information to help improve the understanding of the topic.

Firstly, let's define some terms for those who may not be familiar with them. A field is a mathematical structure where addition, subtraction, multiplication, and division are well-defined operations. Examples of fields include the rational numbers, real numbers, and complex numbers. A polynomial is a mathematical expression consisting of variables and coefficients, with addition, subtraction, and multiplication as its operations. A polynomial ring is a mathematical structure that contains all possible polynomials with coefficients from a specified field.

Now, let's address the question at hand. The statement says that F[x]/(f(x)) is isomorphic to F, meaning that the polynomial ring F[x] modulo the ideal generated by f(x) is equivalent to the field F. This is true if and only if f(x) is a monic irreducible polynomial in F[x]. This means that f(x) cannot be factored into polynomials of lower degree with coefficients in F. In this case, f(x)=x-1 is a monic irreducible polynomial in F[x], and thus, F[x]/(x-1) is isomorphic to F.

Next, the question asks if g(x)=x^2-1 is maximal. In order to answer this, we need to define what a maximal ideal is. An ideal is a subset of a ring (in this case, the polynomial ring F[x]) that satisfies certain properties. A maximal ideal is an ideal that is not contained within any other proper ideal. In other words, if we add any element outside of the ideal to the ideal, it will no longer be an ideal.

In this case, the ideal (x^2-1) is not maximal. It is a proper ideal, but it is contained within the ideal (x-1). This can be seen by the fact that (x^2-1)=(x-1)(x+1), and both (x-1) and (x+1) are polynomials with coefficients in F. Therefore, (x^2-1) is not a maximal ideal.

In conclusion, g(x)=x^2-1 is not a maximal ideal in F[x], but it is a proper ideal. This is because it is contained within the ideal (x-1),