rational integers

[R.D. Silverman]

Originally Posted by R.D. Silverman

Agreed. It does depend on context.

In one context a distinction is made between an algebraic integer defined as an element of a field (or ring)
whose norm is an integer and an actual element of Z, defined as a rational integer. I suspect that the OP
is using it in this context.

[wildrabbitt]

Thanks for replies guys.

With regards to this :

In the book Algebraic Number Theory and Fermat's Last Theorem by Ian Stewart and David Tall, the authors state in page 45 that rational integers are the elements of Z, and integers without any qualifier are algebraic integers, i.e., elements of B.

...because of this (part of a) sentence :

integers without any qualifier

I supposed that there are a number of different types of qualifier an integer can have.

Hence I was surprised after asking "What is a qualifier for an integer?"

..surprised by the answer

An integer is any number whose fractional part is 0.

Now I know I'm assuming that this was an answer to my question and not something else but I'll explain what I thought
due to my assumption :

From the answer I concluded that Stargate the poster, was answering my question by defining what an integer is and therefore

A qualifier for an integer is something that makes it live up to the definition, in other words there is only one qualifier for
an integer, the definition.

Now, I realise that perhaps I'm giving this issue more intellectual attention than it deserves, but to me "qualifier" sounded

like something I'd never learnt before so it seemed mysterious.

Can anyone give a clear explanation of the things I've written about in this post which I'm confused about?

[xilman]

Originally Posted by wildrabbitt

I supposed that there are a number of different types of qualifier an integer can have.

You supposed correctly.

Can anyone give a clear explanation of the things I've written about in this post which I'm confused about?

I'm sure someone can. I'm not sure that I'm the person who can give an answer which will simultaneously satisfy both you and the (quite properly) pedantic.

AFAICT, stargate38 defines an integer as either a counting number (1, 2, 3, 4, ...), zero, or a negative counting number. Each of these has a precise definition and a conventional designation. I'll forego the details and call the entire set of such things Z, which happens to be the conventional designation of that set.

Right, given Z, what else may be of interest? Amongst many choices, we could add the imaginary unit, i into the mix and define (a + b i), where each of a and b are members of the set Z. These beasties are the Gaussian integers which are a special case of algebraic integers. Note that i is a root of the polynomial equation x²+1 = 0, where the coefficients of x are elements of Z --- in this case +1 and +1, and the degree of the polynomial (the highest power appearing) is 2.

Now let the degree of a polynomial p be any value contained within the set Z which is greater than zero and require all coefficients of p also to be elements of Z. Pick any root of p (i.e. a value α such that p(α) = 0). Then the quantity a + α b is defined as an algebraic integer.

[wildrabbitt]

thanks

[xilman]

Originally Posted by wildrabbitt

thanks

You're welcome.

Note that despite a few subsequent edits my definition of an algebraic integer will not satisfy the purists. I'll leave it to them, if any of them are paying attention and can be bothered, to clean up the definition and phrase it in a manner which you find it relatively easy to understand.

(Hint, the word irreducible plays a part. To see why, think about the polynomial p = x² - 4 and its roots.)

[Nick]

Originally Posted by xilman

You're welcome.

Note that despite a few subsequent edits my definition of an algebraic integer will not satisfy the purists. I'll leave it to them, if any of them are paying attention and can be bothered, to clean up the definition and phrase it in a manner which you find it relatively easy to understand.

(Hint, the word irreducible plays a part. To see why, think about the polynomial p = x² - 4 and its roots.)

In my opinion, "integers without any qualifier" was simply intended to mean the word "integers" without any adjective such as "algebraic" (or "Gaussian" or anything else) being placed before it.

In the above description, it is important to note that the polynomials are monic, i.e. the coefficient of the highest power of x is 1. For example, any rational number which is a zero (also called a root) of a monic polynomial with integers as its coefficients must itself be an integer (try and prove this if you have never seen it). This explains why the phrase "rational integers" gets used for ordinary integers: they form the ring of integers of the field of rational numbers.

[Nick]

Just to avoid any misunderstanding: my post above quotes xilman but was addressed to the OP'er.