How Does Proposition 7 Support Proposition 15 in Dummit and Foote?

In summary, Proposition 15 states that the maximal ideals in F[x] are the ideals generated by irreducible polynomials. This follows from Proposition 7 of Section 8.2 in Dummit and Foote, which states that every nonzero prime ideal in a Principal Ideal Domain is a maximal ideal. In one direction, if f(x) is reducible, then F[x]/(f(x)) is not a field. In the other direction, if f(x) is irreducible, then F[x]/(f(x)) is a field.
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I am reading Dummit and Foote Section 9.5 Polynomial Rings Over Fields II and need some help and guidance with the proof of Proposition 15.

Proposition 15 reads as follow:

Proposition 15. The maximal ideals in F[x] are the ideals (f(x)) generated by irreducible polynomials. In particular, F[x]/(f(x)) is a field if and only if f(x) is irreducible.

Dummit and Foote give the proof as follows:

Proof: This follows from Proposition 7 of Section 8.2 applied to the Principal Ideal Domain F[x].


My problem
- can someone show me how Proposition 15 above follows from Proposition 7 of Section 8.2 (see below for Proposition 7 of Section 8.2)

I would be grateful for any help or guidance in this matter.

PeterDummit and Foote - Section 8.2 - Proposition 7

Proposition 7. Every nonzero prime ideal in a Principal Ideal Domain is a maximal ideal.[This problem has also been posted on MHF]
 
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Peter said:
I am reading Dummit and Foote Section 9.5 Polynomial Rings Over Fields II and need some help and guidance with the proof of Proposition 15.

Proposition 15 reads as follow:

Proposition 15. The maximal ideals in F[x] are the ideals (f(x)) generated by irreducible polynomials. In particular, F[x]/(f(x)) is a field if and only if f(x) is irreducible.

Dummit and Foote give the proof as follows:

Proof: This follows from Proposition 7 of Section 8.2 applied to the Principal Ideal Domain F[x].


My problem
- can someone show me how Proposition 15 above follows from Proposition 7 of Section 8.2 (see below for Proposition 7 of Section 8.2)

I would be grateful for any help or guidance in this matter.

PeterDummit and Foote - Section 8.2 - Proposition 7

Proposition 7. Every nonzero prime ideal in a Principal Ideal Domain is a maximal ideal.[This problem has also been posted on MHF]

So in one direction, the conditional is obvious. That is,

Suppose f(x) is reducible:

Then f(x) = g(x)*h(x) for polynomials g and h of degree less than f, which means that
$$
(g(x) + \langle f(x) \rangle)\,(h(x) + \langle f(x) \rangle)
=(f(x) + \langle f(x) \rangle)
=(0 + \langle f(x) \rangle)
$$
Which means that $F[x]/\langle f(x) \rangle$ is not a domain, and therefore not a field.
Now for the other direction, we'll need that proposition:

Suppose f(x) is irreducible:

Because f(x) is irreducible, we know that $\langle f(x) \rangle$ is a prime ideal.
By proposition 7, since F[x] is a principal ideal domain, we know that $\langle f(x) \rangle$ is a maximal ideal.
Finally, since $\langle f(x) \rangle$ is a maximal ideal and F[x]≠{0} is a commutative ring, we know that $F[x]/\langle f(x) \rangle$ is a field.

Thus, the proof is complete.
 

FAQ: How Does Proposition 7 Support Proposition 15 in Dummit and Foote?

What are polynomial rings over fields?

Polynomial rings over fields are algebraic structures in which polynomials with coefficients from a specific field are considered as the elements. These rings are important in abstract algebra and are used in various branches of mathematics.

What is a field?

A field is a mathematical structure in which addition, subtraction, multiplication, and division operations are defined and follow certain rules. Examples of fields include the real numbers, complex numbers, and rational numbers.

How are polynomial rings over fields constructed?

Polynomial rings over fields are constructed by taking a field and adding an indeterminate, usually denoted as x, to it. The elements of the polynomial ring are then formed by combining the indeterminate with coefficients from the field using addition and multiplication operations.

What are the properties of polynomial rings over fields?

Some important properties of polynomial rings over fields include the commutative property, the distributive property, and the existence of a multiplicative identity. Additionally, every nonzero element in a polynomial ring over a field has a multiplicative inverse.

How are polynomial rings over fields used in mathematics?

Polynomial rings over fields are used in various areas of mathematics, including abstract algebra, algebraic geometry, and number theory. They are also important in computer science and engineering, particularly in the study of error-correcting codes and cryptography.

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