Purchase Solution

Sqrarefree Integers, Fields, Conductors and Maximal Ideals

Not what you're looking for?

Ask Custom Question

Let D be a squarefree integer, and let 0 be the ring of integers in the quadratic field Q(sqrtD). For any positive integer f prove that the set Of = Z[fw] = {a + bfw | a, b E Z} is a subring of 0 containing the identity. Prove that [O:Of]= f (index as additive abelian groups). Prove conversely that a subring of 0 containing the identity and having finite index f in 0 (as additive abelian group) is equal to Of. (The ring Of is called the order of conductor f in the field Q(sqrtD). The ring of integers 0 is called the maximal order in Q(sqrtD).

Attachments
Purchase this Solution
Solution Summary

Sqrarefree Integers, Fields, Conductors and Maximal Ideals are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

Solution Preview

Part A:

Taking two arbitrary elements of O_f,

x = a + bfω and y = c + dfω

we find that both

x + y = (a+c) + (b+d)fω

is in O_f , and

x*y = (ac+Dbd) + (ad+bc)fω

is in O_f.

Therefore O_f is a sub-ring.
Taking a = 1 and b = 0, we see that O_f contains the ...

Purchase this Solution
Free BrainMass Quizzes
Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Probability Quiz

Some questions on probability

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

View More Free Quizzes
Related BrainMass Solutions

  • prime ideals

    Prove directly that every maximal ideal is prime. c. For the ring of integers Z, describe all the prime ideals. Which ones are maximal, which ones are not? d.

  • Maximal Ideals, Residues and Ring Homomorphisms

    But we note that and we know should be an integer. This is a contradiction. Therefore, . Since and , then is a proper subset of . Hence is not a maximal ideal in . Maximal Ideals, Residues and Ring Homomorphisms are investigated.

  • Maximal and Principal Ideals

    463406 Maximal and Principal Ideals Let delta=sqrt(-3) and R=Z[delta]. This is not the ring of integers in the imaginary quadratic number field Q[delta]. Let A be the ideal (2,1+delta).

  • Ideals and Factor Rings : Prime Ideal

    17287 Ideals and Factor Rings: Prime Ideal Problem: Let R be a commutative ring. Show that every maximal ideal of R is prime and if R is finite, show that every prime ideal is maximal. ALSO is every prime ideal of Z(integers) maximal? Why?

  • Maximal ideals

    247559 Maximal ideals With respect to the ideal I=<2,x> in Z[x] I believe this ideal is maximal because one theorem I have read suggests to me that all maximal ideals of Z[x] are in the form where p is prime and f(x) is an element of

  • Ring Homomorphisms, Residues and Distinct Maximal Ideals

    Actually, any ideal generated by an linear polynomial is a maximal ideal of . So has infinite many maximal ideals. Ring Homomorphisms, Residues and Distinct Maximal Ideals are investigated. The solution is detailed and well presented.

  • Ideals : Prime but not Maximal

    37878 Ideals : Prime but not Maximal Does the ring Z (integer) contain any ideals that are prime but not maximal? If so, give an example. If not, give a proof. {0} is prime ideal in Z but not maximal.

  • Rings, Prime Ideals and Maximal Ideals

    Rings, Prime Ideals and Maximal Ideals are investigated. The solution is detailed and well presented.

  • Maximal Ideals

    Maximal Ideals are investigated.

View More Related Solutions