We completely classify all polynomials of type

The renowned Rabinowitsch result for complex quadratic fields proved in 1913, published in [

If

See [

A version of Theorem

We will be discussing continued fraction expansions herein for which we remind the reader of the following, the details and background of which may be found in [

We denote the infinite simple continued fraction expansion of a given

A real number

A real number

Now, given a quadratic irrational

Since we are primarily concerned with the case

We need to link quadratic irrationals associated with discriminant

Let

See [

If

Conversely, suppose that we have a quadratic irrational of the form (

It is possible to have a quadratic irrational of type (

Also, to see why we must specialize to quadratic irrationals of type (

Remark

To each quadratic irrational

The infinite simple continued fraction of

We will need the following facts concerning period length.

If

Let

The next result sets the stage for our primary discussion.

Let

See [

Note that the notion of reduction for quadratic irrationals translates to ideals, namely we have the following.

An

To see how this is tied to Definition

See [

If

Now, we let

Let

See [

A reduced

See [

The infrastructure given in Theorem

By Remark

If

If

We begin by stating a very palatable result by Biro that we will employ in our classification.

If

See [

The only values for which

In what follows,

If

First we show that

Now we prove that

If

In the continued fraction expansion of

Suppose that

If

If

Assume to the contrary that

By Theorem

By Lemma

The following is the affirmative solution of four conjectures by this author posed in 1988 in [

For a prime

Suppose that

Suppose that

Suppose that

A nice illustration of Corollary

If

See [

If

As well, we will be employing the following equally pleasant result by Byeon, Kim, and Lee, who used methods similar to those of Biro.

If

See [

If

If

If

If

Let

By virtually the same argument as used in the proof of Theorem

Lastly, we may assume that

Putting Theorems

We list the

Note, as well, that although the original Theorem

If

This remark is provided for the sake of completeness and explaining details in extending the results in [

The following deals with wide Richaud-Degert types and captures the balance of the values using the Euler-Rabinowitsch polynomial

In [

If

We have a list of values for Conjecture

If

By the above discussion, we know that the list (see Table

After the writing of this paper Anitha Srinivasan informed me that, in an unpublished manuscript, she has proved Conjecture

The author gratefully acknowledges the support of NSERC Canada Grant no. A8484. Moreover, thanks go to the referee for suggestions that led to the clarification, increased readability, and streamlining of the presentation.