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user21820
user21820
2:10 PM
For an example of an outright wrong article that I just discovered, see ncatlab.org/nlab/show/theory+of+algebraically+closed+fields. The field C(t) where C is the complex number field and t is an indeterminate is obviously not algebraically complete, but the article's definition implies that it is. It is not a typographical error because it is stated twice in totally different but equivalent ways:
> all polynomials with coefficients from the ring of integers of the prime field have a root
and:
> ACF is the countable collection of sentences in the language ℒ[ring] of rings given by: {field axioms}+{(∃x)(a[n]x^n+…+a[0]=0]}, where the tuples a[0..n] range over all finite tuples of integers.
The two errors (found in both phrasings) are:
(1) They claim that all constant polynomials have a root.
(2) They claim that it suffices to axiomatize for integer polynomials, or equivalently polynomials over the prime field. This is clearly false as my example of C(t) shows.
If this webpage can have two errors stated differently but equivalently, one has to wonder about the reliability of everything else on the page about more complicated claims, not to say other webpages on the same site about more difficult topics. Similarly for many other online sites.
 

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