Math 361, Spring 2017, Assignment 7
Carefully state the following theorems (you do not need to prove them):[edit]
- List of prime ideals of Z.
- List of maximal ideals of Z.
- List of prime ideals of F[x].
- List of maximal ideals of F[x].
- Theorem characterizing when the quotient ring F[x]/⟨m⟩ will be a field.
Solve the following problems:[edit]
- Section 27, problems 5, 7, and 9.
- Construct a field with exactly eight elements. Make addition and multiplication tables for your field. Is your field isomorphic to Z8? (Hint: first find an irreducible cubic in Z2[x].)
Questions[edit]
List of prime ideals of Z-If R is a ring with unity 1, then the map ϕ:Z→R given by ϕ(n)=n⋅1 for n∈Z into R.
List of maximal ideals of Z: Could not fine at all, in my notes or book?
List of prime ideals of F[x]:Could not fine at all, in my notes or book?
List of maximal ideals of F[x]: Let R be a commutative ring with unity. Then M is a maximal ideal of R if and only if R/M is a field.
Theorem characterizing when the quotient ring F[x]/⟨m⟩ will be a field: Let p(x) be an irreducible polynomial in F[x]. If p(x) divides r(x)s(x) for r(x), s(x)∈F[x], then either p(x) divides r(x) or p(x) divides s(x).
So let recap my question here is are the theorems that I listed here are they correct? As well as can you tell me where in the book I might find the, List of maximal ideals of Z and the List of prime ideals of F[x]: I would greatly appreciate the help.
- The lists of prime and maximal ideals of Z are Examples 27.10 and 27.12. The list of maximal ideals of F[x] is Theorem 27.25. Finally, Exercise 27.30 states that the prime ideals of F[x] are the maximal ideals, together with ⟨0⟩. -Steven.Jackson (talk) 10:17, 6 April 2017 (EDT)
Solutions[edit]
- List of prime ideals of Z: ⟨0⟩ and ⟨p⟩ for p prime.
- List of maximal ideals of Z: ⟨p⟩ for p prime.
- List of prime ideals of F[x]: ⟨0⟩ and ⟨f⟩ for f irreducible.
- List of maximal ideals of F[x]: ⟨f⟩ for f irreducible.
- Theorem characterizing when the quotient ring F[x]/⟨m⟩ will be a field: F[x]/⟨m⟩ is a field if and only if m is irreducible.