Math 361, Spring 2017, Assignment 7

From cartan.math.umb.edu


Carefully state the following theorems (you do not need to prove them):[edit]

  1. List of prime ideals of Z.
  2. List of maximal ideals of Z.
  3. List of prime ideals of F[x].
  4. List of maximal ideals of F[x].
  5. Theorem characterizing when the quotient ring F[x]/m will be a field.

Solve the following problems:[edit]

  1. Section 27, problems 5, 7, and 9.
  2. 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].)
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Questions[edit]

List of prime ideals of Z-If R is a ring with unity 1, then the map ϕ:ZR given by ϕ(n)=n1 for nZ 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]

  1. List of prime ideals of Z: 0 and p for p prime.
  2. List of maximal ideals of Z: p for p prime.
  3. List of prime ideals of F[x]: 0 and f for f irreducible.
  4. List of maximal ideals of F[x]: f for f irreducible.
  5. 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.