Re: JSH: Even shorter, disproof

On 06-02-2006 14:29, Arturo Magidin wrote:

I guess that I shall soon become one of those who "no longer respond to
JSH". How do you argue with a guy that writes:

In contrast, the rule is that to be an algebraic integer a number
must be a root of a monic polynomial with integer coefficients, which
importantly DOES NOT FOLLOW from simply defining algebraic integers
to be roots of monic polynomials with integer coefficients.

I think that this will become one of those JSH's immortal sentences.


I think I see here what is going on there: there is a or two word
missing. Algebraic integers are defined as roots of monic polynomials
with integer coefficients. From this it follows, through some basic
algebraic results, that an algebraic integer must be the root of a
monic IRREDUCIBLE polynomial with integer coefficients.

So that's where the problem is! I had a different conjecture: that the
problem was that, given a monic polynomial whose coefficients are
algebraic integers, its roots must also be algebraic integers. True, but
not obvious.

This
particular restriction is not immediately contained in the definition,
and some people had, at some time in the not-so-distant past,
unwilling to grant the latter.

I wonder who you might be talking about... :-)

Best regards,

Jose Carlos Santos
.

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