Zaubertrank's profile - activity

I'm trying to find the solutions to the polynomial $y^2=x^3+1$ over $\mathbb{F}_5$. I have constructed the correct polynomial ring, but I don't know what the analogous function to .roots() is for the two variable case. Thanks

Sage allows you to construct finite fields with a root of any irreducible polynomial as a generator. Thus

sage: PF2.<x> = GF(2)[]
sage: f = x^5 + x^2 + 1
sage: F32.<a> = GF(32, modulus=f)
sage: a.minimal_polynomial()
x^5 + x^2 + 1
sage: PF32.<s> = F32[]
sage: PF32.random_element().roots()
[(a^2 + a + 1, 1), (a^4 + a^3 + a^2, 1)]

Thank you!

So I'm trying to use the .roots() command on a polynomial over the quotient ring F_2[x]/x^5 + x^2 + 1, which is a field isomorphic to F_32. But it keeps giving me the following error:

NotImplementedError: root finding with multiplicities for this polynomial not implemented (try the multiplicities=False option)

Is there a way to get this to work?

Thanks

Yes a lambda function was what I had in mind. Could you extrapolate on how to create these python modules and load them into your sage session, I've been unsuccessful so far using the info I've found online.

I have a function in sage I want to save, but I cannot figure out how. I'm running sage on a mac through the terminal. I'm finding stuff online about save_session, or saving to a document.sage file or something but nothing really seems to be working, is there a straightforward way to do this? Thanks.

Edit: Also while I have your attention, say you have a function f, how do you display the contents of f without actually running it?

That works kcrisman, thanks.

Say I have a function which runs a while loop which always evaluates as true and thus keeps running, is there a way to abort this without just closing the terminal all together? Thanks.

Not every method -- i.e. a function which lives inside an object -- has a function form. What I mean is that you can write sqrt(2), because sqrt is a function, and you could also write 2.sqrt(), but not everything is paired up like that.

This holds for simplify too. There is a simplify function, but you can get much tighter control by calling the simplify methods. You can usually look inside an object by hitting TAB. For example:

sage: q = ( sqrt(2) + sqrt(3) ) * ( sqrt(2) - sqrt(3) ) 
sage: q.[HERE I HIT TAB]
q.N                         q.exp_simplify              q.leading_coeff             q.reduce_trig
q.Order                     q.expand                    q.leading_coefficient       q.rename
q.abs                       q.expand_log                q.left                      q.reset_name
q.add                       q.expand_rational           q.left_hand_side            q.rhs
q.add_to_both_sides         q.expand_trig               q.lgamma                    q.right
q.additive_order            q.factor                    q.lhs                       q.right_hand_side
[etc..]

In sage, lots of functionality lives inside objects like this. If you type

sage: q.simp[TAB]
q.simplify            q.simplify_factorial  q.simplify_log        q.simplify_rational   
q.simplify_exp        q.simplify_full       q.simplify_radical    q.simplify_trig

you'll see a bunch of possibilities. If you type

sage: q.simplify_radical?

you can see the docs for it (and two ?? show the code.)

All of that is a long-winded way to bring us here:

sage: q = ( sqrt(2) + sqrt(3) ) * ( sqrt(2) - sqrt(3) ) 
sage: q.simplify_radical()
-1
sage: q.simplify_full()
-1

oh man great answer, I didn't even know about all that functionality, thanks.

For example: How can I get sage to simplify ( sqrt(2) + sqrt(3) ) * ( sqrt(2) - sqrt(3) ) to -1? I've tried simplify() but it wont do it. Thanks.