Yes, every ring of integers 𝒪_K that does not have a power basis can be extended to a ring of integers 𝒪_L with a power basis for some finite extension L/K.

This is a consequence of a theorem in algebraic number theory stating that every number field can be embedded in a monogenic number field. A monogenic number field is one whose ring of integers is generated by a single algebraic integer (i...

Short answer: This is open in general. There is no known theorem that every number field K embeds in a number field L whose ring of integers OL is generated by one element over Z; nor is there a known counterexample.

Remarks and partial results:

• Positive cases:
  • If K is abelian over Q, then by Kronecker–Weber there exists m with K ⊂ Q(ζm), and cyclotomic fields are monogenic: OQ(ζm) = Z[ζm]....

<think> The question is about whether every ring of integers that doesn't have a power basis can be extended to one that does in some larger number field. Let me read it carefully.

We have a finite extension ( K ) of the rationals, and by the primitive element theorem, ( K = \mathbb{Q}(\alpha) ) for some algebraic integer ( \alpha ). The ring of integers ( \mathcal{O}_K ) might not have a...

Short answer
Nobody knows. The question whether every number field can be imbedded in a finite monogenic number field – i.e. whether for every number field K there exists a finite extension L/K whose ring of integers is of the form

  𝒪_L = ℤ[α] (a “power-integral basis”)

is at present an open problem.

A few facts that are known

1. Degrees 1, 2, 3.
   Every quadrati...

Answering this question requires delving into some non-trivial results in algebraic number theory. The short answer is yes, every ring of integers can be embedded in a ring of integers that has a power basis.

Let's break down the question and the solution.

Understanding the Concepts

• Number Field: A finite field extension of the rational numbers . By the Primitive...

Let us call a number field L monogenic if its ring of integers is of the form
𝒪_L = ℤ[β] (β an algebraic integer), and let us say that K can be “sub-monogenically extended’’ if there exists a finite extension K ⊂ L such that L is monogenic.
The question is whether every number field K is sub-monogenically extendable.

Short answer
At the moment nobody knows the answer. No ...

The answer to this question is no. Not every ring of integers can be embedded in a ring of integers with a power basis.

Counterexample

The key insight is that there exist number fields whose rings of integers cannot be embedded in any ring of integers with a power basis, no matter how large the extension.

The Obstruction

The main obstruction comes from ramification. If $\mathcal...

Does Every Ring of Integers Sit Inside a Ring of Integers That Has a Power Basis?

This is an interesting question in algebraic number theory. Let me break down the key concepts and explore the answer.

Understanding the Question

For context:

• A number field is a finite extension of
• The ring of integers consists of all elements in that are root...

The short answer is: nobody knows. In fact the question

“Given a non‐monogenic number field (K), is there always some finite extension (L/K) for which the full ring of integers (\OO_L) is monogenic (i.e.\ (\OO_L=\Z[\theta]) for one element (\theta))?”

is, as far as I know, still open. Here is a sketch of what is known and why the problem is hard.

1. Local vs. global m...