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# integral element

An element $a$ of a field $K$ is an integral element of the field $K$, iff

$|a|\leq 1$ |

for every non-archimedean valuation $|\cdot|$ of this field.

The set $\mathcal{O}$ of all integral elements of $K$ is a subring (in fact, an integral domain) of $K$, because it is the intersection of all valuation rings in $K$.

Examples

1. $K=\mathbb{Q}$. The only non-archimedean valuations of $\mathbb{Q}$ are the $p$-adic valuations $|\cdot|_{p}$ (where $p$ is a rational prime) and the trivial valuation (all values are 1 except the value of 0). The valuation ring $\mathcal{O}_{p}$ of $|\cdot|_{p}$ consists of all so-called p-integral rational numbers whose denominators are not divisible by $p$. The valuation ring of the trivial valuation is, generally, the whole field. Thus, $\mathcal{O}$ is, by definition, the intersection of the $\mathcal{O}_{p}$’s for all $p$; this is the set of rationals whose denominators are not divisible by any prime, which is exactly the set $\mathbb{Z}$ of ordinary integers.

2. If $K$ is a finite field, it has only the trivial valuation. In fact, if $|\cdot|$ is a valuation and $a$ any non-zero element of $K$, then there is a positive integer $m$ such that $a^{m}=1$, and we have $|a|^{m}=|a^{m}|=|1|=1$, and therefore $|a|=1$. Thus, $|\cdot|$ is trivial and $\mathcal{O}=K$. This means that all elements of the field are integral elements.

3. If $K$ is the field $\mathbb{Q}_{p}$ of the $p$-adic numbers, it has only one non-trivial valuation, the $p$-adic valuation, and now the ring $\mathcal{O}$ is its valuation ring, which is the ring of $p$-adic integers; this is visualized in the 2-adic (dyadic) case in the article “$p$-adic canonical form”.

## Mathematics Subject Classification

12E99*no label found*

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## Comments

## question to 'integral element'

When I saw the entry 'integral element' I thought it is connected somehow to integration, and I was quite suprised that it was not the case ;) But nevertheless, is there any connection to integration? And why actually one names such elements as 'integral'?

Thanks in advance.

Regards

Serg.

-------------------------------

knowledge can become a science

only with a help of mathematics

## Re: question to 'integral element'

> When I saw the entry 'integral element' I thought it is

> connected somehow to integration, and I was quite suprised

> that it was not the case ;) But nevertheless, is there any

> connection to integration? And why actually one names such

> elements as 'integral'?

In this sense, the word 'integral' is an adjective describing the property of being an integer, or similar to an integer. It is used commonly in algebra and number theory.

The fact that 'integral' in the sense of analysis and in the sense of number theory is the same word is a historical accident.

## Re: question to 'integral element'

``Integral (element)'' has in mathematics nothing to do with ``integration''. Ethymologically, both words have their origin in the adjective ``integral'', the basic sense of which is 'whole, entire'.

The origin of the English ``integral'' is Latin ``integer, integra, integrum'' having the initial meaning 'in-tact, un-touched, un-broken, whole' (the Latin prefix ``in'' means 'non', ``teger'' is a derivative of the verb ``tangere'' = 'to touch').

Isn't the meaning of the integration to form something "entire", viz. the antiderivative, from its "shadow", viz. the derivative?

Jussi

## Re: question to 'integral element'

The word "integral in the sense of calculus indeed means "whole" or "entire". The original name for differential calculus was "calculus differentialis", which I think means "calculus of differences". At first Leibniz used the term "calculus summatorius" for integral calculus, which I guess means "calculus of summations". Later Jacob Bernoulli came up with the phrase "calculus integralis", which means "calculus of the whole" as opposed to only the differences. This terminology is the one that stuck.

## another questions to 'integral element'

Thanks a lot! Really nice and full explanations!!!

I am actually quite suprised that "integral" and "integer" are in fact related. But your explanations is a perfect motivation for it:

> The origin of the English ``integral'' is

> Latin ``integer, integra, integrum''

> having the initial meaning 'in-tact, un-touched, un-broken, whole'

But that

> It is used commonly in algebra and number theory

I didn't know. Although, I have just remembered that in the entry "integral domain" this misleading in terminology is discussed ;)

But then let me ask another things: why "non-archimedean valuation" actually are called non-archimedian and why they lead to integers. Of cource the last question is connected with the fact that

"The only non-archimedean valuations of $\mathbb{Q}$ are the $p$-adic valuations"

but for me this fact is not obvious. Is there probably some intuitive motivation for it?

Thanks in advance!

Regards

Serg.

-------------------------------

knowledge can become a science

only with a help of mathematics

## linker bug?

I have noticed the following thing in the above entry ("integral element"):

there is an expression "$p$-adic valuations", BUT it doesn't get linked to the entry "$p$-adic valuation".

It seems it is connected not with plural form of valuations, since linker doesn't care about plural/possesive, but with math environment $$. May be it is something to do with the following: when one copys the above expression one gets additional space in $$, i.e.:

$ p$-adic valuations

So, may be linker sees the expression with additional space and that's why doesn't link it?

Serg.

-------------------------------

knowledge can become a science

only with a help of mathematics

## Re: another questions to 'integral element'

Please see the Theorem 3 in "ultrametric triangle inequality"; the usual absolute value (e.g. in Q) has the Archimedean property, but no p-adic valuation (e.g. in Q) has it.

Regards,

Jussi