I may be confused, but...

[Justin C. Walker]

Using sage 9.7 and 10.3 (on an oldish iMac), I ran into some puzzling results, of which the following are minimal examples:

#1:

sage: 4.is_square()

True

sage: 4.is_squarefree()

False

#2:

sage: f=x^2-3

sage: K.<a>=NumberField(f)

sage: OK=K.maximal_order()

sage: v=OK.random_element()

sage: v

2*a + 1

sage: v==v.conjugate()

True

#1 I don’t get (and it seems to contradict the “?” doc); while #2 may be a misunderstanding on my part.

Thoughts?  Pointers?

Thanks, for any help.

Justin

--

Justin C. Walker, Curmudgeon at Large

Institute for the Absorption of Federal Funds

-----------

I want to die, peacefully in my sleep, like my grandfather;

not screaming in terror, like his passengers.

[John Cremona]

As for #1, 4 is certainly a square, and is also not square-free (begin divisible by a square >1) so I don't see any problem.

For #2, K is a real quadratic field, and the conjugate method is *not* Galois conjugation, but complex conjugation (with respect to either of the embeddings of K into CC); for that, use v.galois_conjugate().

John Cremona

[Justin C. Walker]

Thanks for that lesson in the English language, John.  I obviously needed it :-}

As for #2, that’s something I did not know!

Justin

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Justin C. Walker, Curmudgeon at Large

Director

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Question 43:

What if the hokey pokey

really *is* what it’s all about?

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[Nils Bruin]

Note that

4.is_square?

and

a=4

a.is_square?

give different documentation strings. I don't know where the first is coming from. That looks like a strange glitch. It gives the documentation of the function is_square, which is not what the method on integers is.