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I'm trying to work with a quotingring of a matrix space. Now I use the following code. But since $n$ is my modulus I expect my second matrix to be equal to $M_1$ in this quotient ring. But it's like its just ignoring the fact that its a quotient ring and it does every computation in MS. What am I doing wrong or is my understanding of quotient rings flawed?

Thanks in advance.

MS = MatrixSpace(ZZ,2,2)

n=MS([[17,19],[11,31]])

R = MS.quotient_ring(MS*n*MS) #or R = MS.quotient_ring(n)
M1 = R([[1,1],[1,1]])

print(M1)

M2 = R(n+M1.lift())

print(M2)

this code gives this as a result

[1 1]
[1 1]

[18 20]
[12 32]

I'm trying to work with a quotingring of a matrix space. Now I use the following code. But since $n$ is my modulus I expect my second matrix to be equal to $M_1$ in this quotient ring. But it's like its just ignoring the fact that its a quotient ring and it does every computation in MS. What am I doing wrong or is my understanding of quotient rings flawed?

Thanks in advance.

MS = MatrixSpace(ZZ,2,2)

n=MS([[17,19],[11,31]])

R = MS.quotient_ring(MS*n*MS) #or R = MS.quotient_ring(n)
M1 = R([[1,1],[1,1]])

print(M1)

M2 = R(n+M1.lift())

print(M2)

p=5
F.<c>=GF(p^p,modulus=x^p-x-1)
V=[c^i for i in [0..p-1]]
s=c^6+c^3+c^2+1

Now I want to get the coeffecients of s represented to the base V[i], how do i do that

I have a linear map $\alpha$ from $F_{p^n} \longrightarrow F_{p^n}$, where we see $F_{p^n}$ as a vector space over $F_p$ with a $V_i$ as base elements. I want to create the matrix representation for $\alpha$. For that I have to calculate $\alpha(V_i)$ and then write it in the basis $V_i$ to get my values for the matrix. How exactly do I do the last in sage ? For eg. a polynomial ring it's easy because the elements are already written according to its base but in general thats not the case...

I'm trying to construct some field extenstions of GF(p), this is what I have

p=5
F=GF(p)
R1.<x> = F['x']
F1.<alpha1> = F.extension(x^p - x - 1)
R2.<x> = F1['x']
F2.<alpha2> = F1.extension(x^p - x - alpha1^(p-1))
R3.<x> = F2['x']
F3.<alpha3> = F2.extension(x^p - x - (alpha1*alpha2)^(p-1))
R4.<x> = F3['x']
F4.<alpha4> = F3.extension(x^p - x - (alpha1*alpha2*alpha3)^(p-1))

It creates F3 like it should but it doesn't create F4. I get a NotImplementedError... What did I do wrong?

I have to construct a chain like that, with 5 different elements. When i use the extend function for the 4th time it gives "NotImplementedError"...

I would like to construct the field Fp(alpha,beta) where alpha is a root of x^p-x-1 (over Fp[x]) and beta is a root of the polynomial x^p-x-alpha^(p-1) (over Fp(alpha)[x]).

I have tried the following

 F0.<x>=GF(p)['x']
 f1=x^p-x-1
 R1.<alpha1>=F0.quotient(f1)['alpha1']
 F1.<x>=Frac(R1)

 f2=x^p-x-alpha1^(p-1)
 R2.<alpha2>=F1.quotient(f2)['alpha2']
 F2.<x>=Frac(R2)

but I think it creates the field F1 well, but it goes wrong for R2... It also feels like there should be a much more straightforward way to do this in sage. What would be the proper way to do this ?

I use this piece of code in my project:

print h
print h(3.2)
print parent(h)
plot(h(x) , (-4,4), thickness=2, color='green' )

and this is my output:

2.14250281996159*x + 1.74284059736793
8.59884962124501
Univariate Polynomial Ring in x over Real Field with 53 bits of precision
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_231.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cHJpbnQgaApwcmludCBoKDMuMikKcHJpbnQgcGFyZW50KGgpCnBsb3QoaCh4KSAsICgtNCw0KSwgdGhpY2tuZXNzPTIsIGNvbG9yPSdncmVlbicgKQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>

  File "/var/sage/tmp2yTHkg/___code___.py", line 6, in <module>
    exec compile(u"plot(h(x) , (-_sage_const_4 ,_sage_const_4 ), thickness=_sage_const_2 , color='green' )" + '\n', '', 'single')
  File "", line 1, in <module>

  File "/opt/sage/sage-6.3/local/lib/python2.7/site-packages/sage/misc/decorators.py", line 705, in wrapper
    return func(*args, **kwds)
  File "/opt/sage/sage-6.3/local/lib/python2.7/site-packages/sage/misc/decorators.py", line 550, in wrapper
    return func(*args, **options)
  File "/opt/sage/sage-6.3/local/lib/python2.7/site-packages/sage/plot/plot.py", line 1163, in plot
    G = funcs.plot(*args, **original_opts)
  File "polynomial_element.pyx", line 286, in sage.rings.polynomial.polynomial_element.Polynomial.plot (build/cythonized/sage/rings/polynomial/polynomial_element.c:5907)
AttributeError: must give both plot endpoints

Why does this happen ? I have specified my endpoints right? so why is it saying I didn't ? I tried a lot to solve this but i can't get it solved.. anyone who can tell me whats wrong ?

Thanks in advance

I have this equation : sqrt((b-a)^2 + (B-A)^2) == A+B and i would like to have it solved to b-a=+- 2 * sqrt(AB) in sage. Right now I have the following code but it doesn't really output what I want. I just get
A^2 - 2AB + B^2 + a^2 - 2ab + b^2 == A^2 + 2AB + B^2
and I don't know how to make sage solve it further.

my code:

var('a,b,c,A,B,C')
eq1 = (sqrt((b-a)^2 + (B-A)^2) == A+B)
(eq1^2).expand()

Thanks in advance