ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 20 May 2024 10:35:52 +0200exp(0) and type conversion in a p-adic extensionhttps://ask.sagemath.org/question/77527/exp0-and-type-conversion-in-a-p-adic-extension/I'm trying to use a $p$-adic extension, and I think I have found two bugs.
The first one: When I try to calculate $exp(0)$, the result is an error instead of 1.
R.<a>=Zq(9,prec=5)
zero=R.zero()
exp(zero)
ValueError: unit part of 0 not defined
The second one happens when I try to convert from R to the base ring. I am using zero in the example, but any value (in the base ring) gives the same problem.
R.<a>=Zq(9,prec=5)
zero=R.zero()
R0=R.base_ring()
R0(zero)
RecursionError: maximum recursion depth exceeded in comparison
I think these are bugs because the operations seem completely valid. This is Sage 10.2.Luis CrespoMon, 20 May 2024 10:35:52 +0200https://ask.sagemath.org/question/77527/Modular Formshttps://ask.sagemath.org/question/76300/modular-forms/What is the most efficient way to determine the coefficients $$a_{1}, a_{2}, ..., a_{21}$$ such that
$$\Delta^{21} \mid T_{3} \equiv \sum_{i = 1}^{21} a_{i} \Delta^{i} \ (\text{mod }2)\ ?$$
I tried finding the $q$-series expansion of LHS and accordingly kept subtracting powers of $\Delta$ to find the scalars, but is there a better way to do this?
By $\Delta(z)$, I mean the Ramanujan Delta function which is the $24$th power of Dedekind eta function.
For a prime $p$ and $z \in \mathbb{H}$, we have
$(f \mid T_{p})(z) = (T_{p}(f))(z) = p^{k - 1} f(pz) + \frac{1}{p} \:\displaystyle{\sum_{b = 0}^{p - 1}} \: f \left(\frac{z + b}{p} \right)$.
Furthermore, in my case,
$f(z) = \Delta(z) = q \: \displaystyle{\prod_{n \geq 1} (1 - q^{n})^{24}}$ where $q = e^{2 \pi i z}$ and $z \in \mathbb{H}$.SwatiFri, 01 Mar 2024 20:16:30 +0100https://ask.sagemath.org/question/76300/How to write a p-adic exponent b^k as a Power series in k ?https://ask.sagemath.org/question/65075/how-to-write-a-p-adic-exponent-bk-as-a-power-series-in-k/Let $b$ be p-adic number, we write $b$ as a Power series in $p$ with a given precison. Is It possible to write $b^k$ as a Power series in $k$, with $k$ an integer ? An example : Let $\gamma_1$ and $\gamma_2$ be the 3-adic unit roots of the quadratic equations $x^2+x+3=0$ and $x^2+2x+3=0$ respectively. Let $k$ be an integer. Let
$$c(k) = \frac{\gamma_1^k}{\gamma_1^2 -3} + \frac{\gamma_2^k}{\gamma_2^2 -3} + 1$$
The problem is to show that
$$v_3(c(k) - 9(-1+4k^2)-27(k^3 + k^4))\geq 4.\qquad (1)$$
I know how to write $\gamma_1$ and $\gamma_2$ as a 3-adic power series with given precision, but have no idea how to work with the exponent. I've tried (1) for k integer between 1 and 100 and the inequality is true only for even numbers. For odd numbers, the left side of (1) is zero.
The inequality (1) is from the article Numerical experiments on families of modular forms by Coleman, Stevens, and Teitelbaum, page 7.ClaudiodsvSat, 26 Nov 2022 13:51:22 +0100https://ask.sagemath.org/question/65075/Can I create 10-adic numbers?https://ask.sagemath.org/question/63064/can-i-create-10-adic-numbers/I'm learning about P-adics, and would like to play around a bit with 10-adic numbers, to follow along with some tutorials.
Unfortunately so far all ways I've tried to create 10-adic numbers in Sage resulted in the error "p must be prime".
I understand non-prime-adics have their issues, but can I still make it work somehow?Lilly2Wed, 29 Jun 2022 22:29:15 +0200https://ask.sagemath.org/question/63064/Creating a polynomial from a string with symbolic constantshttps://ask.sagemath.org/question/63059/creating-a-polynomial-from-a-string-with-symbolic-constants/Below is some code that does the following. First, define `R` to be the univariate polynomial ring `Q_2[x]`, and let `f(x)` be the polynomial `x^2 - 2` in this ring. Then let `K` be the totally ramified quadratic extension defined by this polynomial, and call its generator a. Let `S` be the polynomial ring `K[y]`. Then I would like to define a polynomial `y + a` in `S`. This works find if I write `f1 = S(y+a)`, but it fails if I try `f2 = S('y + a')`. However, I need to be able to define my polynomial from a string as in the `f2` case. Can anyone help?
R.<x> = Qp(2,100)[]
f = R(x^2 - 2)
K.<a> = Qp(2,100).ext(f)
S.<y> = K[]
# The next line works
f1 = S(y + a)
# The next one throws an error
f2 = S('y + a')Sebastian MonnetWed, 29 Jun 2022 18:32:42 +0200https://ask.sagemath.org/question/63059/LMFDB Code for p-adic extensions gives NotImplementedErrorhttps://ask.sagemath.org/question/63042/lmfdb-code-for-p-adic-extensions-gives-notimplementederror/I downloaded the Sage code from [this](https://www.lmfdb.org/padicField/?p=2&n=4) page, which is supposed to give me all quartic extensions of the 2-adic integers. When I try to run the code, it gives me the error:
NotImplementedError: Extensions by general polynomials not yet supported. Please use an unramified or Eisenstein polynomial.
A mwe for the error is given by the following code:
pAdicExtension(Qp(2, 100), PolynomialRing(Qp(2, 100),'x')([4, 0, 8, 0, 1]),var_name='x')
I am using SageMath 9.6, which I believe is the latest version, so it seems strange that code from the LMFDB would be giving me this error, since presumably they tested it before putting it up. Can anyone advise me on getting around this or fixing it?Sebastian MonnetTue, 28 Jun 2022 13:54:05 +0200https://ask.sagemath.org/question/63042/Extension field over p-adics: how to write an element in the standard basis?https://ask.sagemath.org/question/56008/extension-field-over-p-adics-how-to-write-an-element-in-the-standard-basis/Suppose we have an extension field $\mathbb{Q}_2(w)$, where $w$ is a root of $f(x) = x^3 + 4x^2 + 2$. By default, Sage represents $w^3$ as $w^3 + O(w^d)$, where $d$ is the precision. How do I get Sage to print $w^3$ out as a linear combination of the standard basis, i.e., as $-4w^2 - 2$ (with -4 and -2 written as they would be in $\mathbb{Q}_2$)?caelanritterWed, 03 Mar 2021 20:50:03 +0100https://ask.sagemath.org/question/56008/How can I compute a fixed field over the p-adicshttps://ask.sagemath.org/question/55336/how-can-i-compute-a-fixed-field-over-the-p-adics/I have problems to implement the following set up:
I want to have a field $K = \mathbb{Q}_3$ and an extension $L = \mathbb{Q}_3(\alpha)$ over $K$ where $f:=\min_K(\alpha) = x^4 - 3x^2 + 18$. This extension has degree $4$ and ramification index $2$. Furthermore, let $F/K$ be the unique unramified extension of $K$ of degree $4$ which is generated by a primitive $5$-th root of unity $\zeta_5$.
Then one can show that $\varphi: \alpha \mapsto \frac{(2 \alpha^2 - 3)\sqrt{-\frac{2}{7}}}{\alpha},$ $\zeta_5 \mapsto \zeta_5^3$ is an element of the Galois group of $LF/K$. Now let $L' = (LF)^{\langle \varphi \rangle}$. This must be a quadratic and totally ramified extension of $K$. There are only two possibilities for that: $K(\sqrt{3})$ or $K(\sqrt{-3})$.
**Question**: How to determine whether $L' = K(\sqrt{3})$ or $L' = K(\sqrt{-3})$ (or equivalently, $\varphi(\sqrt{3}) = \sqrt{3}$ or $\varphi(\sqrt{-3}) = \sqrt{-3}$)?
Since I only have only superficial knowledge about Sage, I was not even able to set up the easy things like the field $L$ properly. When I use
K = Qp(3)
R.<x> = ZZ[]
f = x^4 - 3*x^2 + 18
L.<alpha> = K.extension(f)
I get an error that my polynomial $f$ must be either unramified or Eisenstein (which of course does not exist since $L/K$ is neither unramified nor totally ramified). Furthermore, I have no idea how to approach with my problem with Sage otherwise. And since computation by hand is pretty hard in this case (I already tried!), it would be nice to solve with problem here, so I can use it for similar computations in the future.
Could you please help me with this problem?RotdatMon, 18 Jan 2021 16:41:46 +0100https://ask.sagemath.org/question/55336/$p$-adic extension of $n$th root of unity.https://ask.sagemath.org/question/47812/p-adic-extension-of-nth-root-of-unity/I have used the following command to define the 5-adic Unramified extension ring in c defined by the polynomial $x^3 + 3x + 3$:
Sage: R.<c> = zq(125, prec=20)
Now, I want to find all the $n$th root of unity in this ring for $n$ dividing $124$. I dont know, how the $n$-th roots are implemented. Kindly help me with this.
Thank you.GA316Tue, 10 Sep 2019 11:41:54 +0200https://ask.sagemath.org/question/47812/How to define $\mathbb{Q}_p(\sqrt{5})$ and $\mathbb{Q}_p (\sqrt{5} ,\sqrt{3})$ and find their valuation rings for $p=7$?https://ask.sagemath.org/question/45695/how-to-define-mathbbq_psqrt5-and-mathbbq_p-sqrt5-sqrt3-and-find-their-valuation-rings-for-p7/I tried finite extension $\mathbb({Q}_p$ I unable do it. It will also great if help me with how to define the valuation ring of that finite extensionSunil pasupulatiThu, 07 Mar 2019 05:03:29 +0100https://ask.sagemath.org/question/45695/i want to find factorization ideal (3) in integral closure of Z_3 in Q_3(sqrt(2),sqrt(3))https://ask.sagemath.org/question/44067/i-want-to-find-factorization-ideal-3-in-integral-closure-of-z_3-in-q_3sqrt2sqrt3/ my problem is to define Q_3(sqrt(2),sqrt(3))
ii)find factorization of ideal
Sunil pasupulatiThu, 25 Oct 2018 10:48:56 +0200https://ask.sagemath.org/question/44067/Algorithm for finding a defining polynomial for an unramified extension?https://ask.sagemath.org/question/42575/algorithm-for-finding-a-defining-polynomial-for-an-unramified-extension/ I wanted to know by what algorithm sage finds a defining polynomial of an unramified extension of p-adic numbers?mathjainTue, 12 Jun 2018 22:38:01 +0200https://ask.sagemath.org/question/42575/programming of looping to print selected value of mhttps://ask.sagemath.org/question/35264/programming-of-looping-to-print-selected-value-of-m/I have the following code for m=1:
Qp=pAdicField(13)
E=EllipticCurve(Qp,[0,0,0,-3267,45630]); E
P=E([-21,324]);P
p=13;p #because I am working 13-adically
R=12104*1*P;R #m=1
S=-2*P;S
Q=R+S;Q
(12 + 8*13 + 5*13^2 + 10*13^3 + 10*13^4 + O(13^5) : 9 + 3*13 + 6*13^2 + 13^3 + 5*13^4 + 2*13^5 + O(13^5) : 1 + O(13^5))
x=Q[0];x
12 + 8*13 + 5*13^2 + 10*13^3 + 10*13^4 + O(13^5)
y=Q[1];y
9 + 3*13 + 6*13^2 + 13^3 + 5*13^4 + O(13^5)
W=(81*x^16 + 40662*x^15 + 14353281*x^14 - 460241028*x^13 - 644722959186*x^12 + 39379675354740*x^11 + 5212980804862026*x^10 - 415546630058854656*x^9 - 8202010485984353739*x^8 + 1396767997483732402758*x^7 - 27550698906220673513787*x^6 - 1044392234943529703379852*x^5 + 60770398462922893831446348*x^4 - 1284453663719469166478575296*x^3 + 14183844641879715988450074288*x^2 - 81800517874945025246941522368*x + 196162341839727571433321441856)- (3240*x^14 + 456840*x^13 + 188268624*x^12 - 45834271200*x^11 - 2435651997264*x^10 + 682353767281968*x^9 - 7053953405575680*x^8 - 2553415737499629216*x^7 + 98906717445152189544*x^6 + 1348117411901578667784*x^5 - 162666175355778441465360*x^4 + 4276857451171442758058304*x^3 - 54456600108308451946891776*x^2 + 350065581968511893813480064*x - 918312303919436410092339456)*y;W
2 + 4*13 + 3*13^2 + 11*13^3 + 11*13^4 + O(13^5)
D=W[0];D
2
T=kronecker(D,p);T
-1
So in this example I reject m=1 because T=-1. So basically I want to do a loop for m=1,2,...,10, and everytime T=-1 I will reject that value of m and only leave set of numbers of m=[2,..10] that gives T=1. I tried doing the coding to work but it keeps giving error regarding the "m" value. I did something like this
Qp=pAdicField(13)
E=EllipticCurve(Qp,[0,0,0,-3267,45630]); E
P=E([-21,324]);P
p=13;p #because I am working 13-adically
R=12104*m*P;R #I use a general m which I will define later that m=1,2,...,10
S=-2*P;S
Q=R+S;Q
x=Q[0];x
y=Q[1];y
W=(81*x^16 + 40662*x^15 + 14353281*x^14 - 460241028*x^13 - 644722959186*x^12 + 39379675354740*x^11 + 5212980804862026*x^10 - 415546630058854656*x^9 - 8202010485984353739*x^8 + 1396767997483732402758*x^7 - 27550698906220673513787*x^6 - 1044392234943529703379852*x^5 + 60770398462922893831446348*x^4 - 1284453663719469166478575296*x^3 + 14183844641879715988450074288*x^2 - 81800517874945025246941522368*x + 196162341839727571433321441856)- (3240*x^14 + 456840*x^13 + 188268624*x^12 - 45834271200*x^11 - 2435651997264*x^10 + 682353767281968*x^9 - 7053953405575680*x^8 - 2553415737499629216*x^7 + 98906717445152189544*x^6 + 1348117411901578667784*x^5 - 162666175355778441465360*x^4 + 4276857451171442758058304*x^3 - 54456600108308451946891776*x^2 + 350065581968511893813480064*x - 918312303919436410092339456)*y;W
D=W[0];D
T=kronecker(D,p);T
for m in range(10) :
if T == 1:
print(m)
Unfortunately, the programming is not working, keeps giving error for `m`.
By the way, I got the following code working on PARI, but I prefer to use SAGE. Is there a way to get the coding to work on SAGE too.
for(m=1,10,R=12104*m*P;p=13;x=R[1];y=R[2]; W=(81*x^16 + 40662*x^15 + 14353281*x^14 - 460241028*x^13 - 644722959186*x^12 + 39379675354740*x^11 + 5212980804862026*x^10 - 415546630058854656*x^9 - 8202010485984353739*x^8 + 1396767997483732402758*x^7 - 27550698906220673513787*x^6 - 1044392234943529703379852*x^5 + 60770398462922893831446348*x^4 - 1284453663719469166478575296*x^3 + 14183844641879715988450074288*x^2 - 81800517874945025246941522368*x + 196162341839727571433321441856)- (3240*x^14 + 456840*x^13 + 188268624*x^12 - 45834271200*x^11 - 2435651997264*x^10 + 682353767281968*x^9 - 7053953405575680*x^8 - 2553415737499629216*x^7 + 98906717445152189544*x^6 + 1348117411901578667784*x^5 - 162666175355778441465360*x^4 + 4276857451171442758058304*x^3 - 54456600108308451946891776*x^2 + 350065581968511893813480064*x - 918312303919436410092339456)*y;q=lift(W);a=q*(denominator(q))^2;if(kronecker(a,p)<0, print1(m "\t")))ShaThu, 27 Oct 2016 12:10:22 +0200https://ask.sagemath.org/question/35264/change p-adic precision in elliptic curvehttps://ask.sagemath.org/question/35262/change-p-adic-precision-in-elliptic-curve/ I have the following code
Qp=pAdicField(13)
E=EllipticCurve(Qp,[0,0,0,-3267,45630]); E
P=E([-21,324]);P
and I get answer for point P
(5 + 11*13 + 12*13^2 + 12*13^3 + 12*13^4 + 12*13^5 + 12*13^6 + 12*13^7 + 12*13^8 + 12*13^9 + 12*13^10 + 12*13^11 + 12*13^12 + 12*13^13 + 12*13^14 + 12*13^15 + 12*13^16 + 12*13^17 + 12*13^18 + 12*13^19 + O(13^20) : 12 + 11*13 + 13^2 + O(13^20) : 1 + O(13^20))
How to change the precision of the power of 13. I am only interested up to O(13^4)ShaThu, 27 Oct 2016 11:31:26 +0200https://ask.sagemath.org/question/35262/Coding p-adic Newton's method to solve polynomials?https://ask.sagemath.org/question/32608/coding-p-adic-newtons-method-to-solve-polynomials/ I'm confused as to how to code the p-adic Newton's method in order to solve polynomials, and specifying solutions to a specific number of p-adic digits. Any help?veryconfusedSat, 20 Feb 2016 20:54:57 +0100https://ask.sagemath.org/question/32608/Can't construct automorphisms of p-adic fieldshttps://ask.sagemath.org/question/29390/cant-construct-automorphisms-of-p-adic-fields/ I'm trying to construct automorphisms of finite extensions of $\mathbb Q_p$ and getting a funny error. Here's a prototypical example:
Evaluating the cyclotomic polynomial $x^4 + x^3 + x^2 + x + 1$ at $x+1$ gives an Eisenstein polynomial for the prime $p=5$. If $\pi$ is a root of $(x+1)^4 + (x+1)^3 + (x+1)^2 + (x+1) + 1$ then $\pi+1$ will be a primitive 5th root of unity.
K.<pi> = Qp(5).ext(sum((1+x)^i for i in range(5)))
So $K=\mathbb Q_p(\zeta_5)$ with uniformizer $\pi=\zeta_5-1$. Now I want to define the automorphism $\tau:\zeta_5\mapsto\zeta_5^2$. We have $\tau(\pi)=(1+\pi)^2-1=2\pi+\pi^2$. But the following
tau = K.hom([2*pi+pi^2])
results in the error `TypeError: images do not define a valid homomorphism`. What's going on? Are `hom`'s of $p$-adic fields not really implemented yet or am I doing something wrong?siggytmThu, 03 Sep 2015 00:40:36 +0200https://ask.sagemath.org/question/29390/Finding p-adic valuations in high degree cyclotomic fieldshttps://ask.sagemath.org/question/26900/finding-p-adic-valuations-in-high-degree-cyclotomic-fields/I'm looking at a cyclotomic field ${\bf Q}(\mu_{p(p-1)})$ for $p$ a prime around 50 and so this field has fairly large degree. In this field, $p$ has ramification index $p$ and has $p-1$ primes sitting above it.
I'm trying to compute the valuation of an element in this field at any of these primes above $p$. Using commands like "primes_above" won't seem to work as the computer just hangs presumably because this extensions degree is just too big.
Questions:
1) Is there another way to compute $p$-adic valuations in this field?
2) Locally, this is only a $p$-th degree extension of ${\bf Q}_p$. So I created a p-adic field by using pAdicField(p).ext(1+(x+1)+(x+1)^2+...+(x+1)^(p-1)) to create this local p-th degree extension of Q_p. However, I can't find any way to map my global elements in ${\bf Q}(\mu_{p(p-1)})$ to this local field. Any ideas on how to proceed along these lines?
Robert PollackWed, 20 May 2015 15:47:43 +0200https://ask.sagemath.org/question/26900/Echelon form of matrix with coefficients in Zphttps://ask.sagemath.org/question/25142/echelon-form-of-matrix-with-coefficients-in-zp/I was wondering if there was a way to find the echelon form of a matrix with coefficients in Zp with sage? I tried
<pre><code>Matrix(Zp(5), 3, 3, [1..9]).echelon_form()</code></pre>
but this gives a NotImplementedError as echelon_form is not implemented over generic non-exact rings at present. I would also very much like the transformation matrix used to get to echelon form if possible.hypercubeFri, 05 Dec 2014 17:10:06 +0100https://ask.sagemath.org/question/25142/how to a p-adic expansion of a rational function in sage?https://ask.sagemath.org/question/23429/how-to-a-p-adic-expansion-of-a-rational-function-in-sage/http://www.maplesoft.com/support/help/Maple/view.aspx?path=padic/orderp
there are more function about p-adic number than maple,but maple has some about p-adic function field,
why not "copy" it?cjshWed, 16 Jul 2014 08:29:12 +0200https://ask.sagemath.org/question/23429/is multiplicative group of Q5 right?https://ask.sagemath.org/question/10574/is-multiplicative-group-of-q5-right/
K = Qp(5, print_mode='digits')
C1=CartesianProduct(ZZ,Integers(4));C1
Cartesian product of Integer Ring, Ring of integers modulo 4
C2=CartesianProduct(C1,Z5);C2
Cartesian product of Cartesian product of Integer Ring, Ring of integers modulo 4, 5-adic Ring with capped relative precision 20
C2.is_ring();cjshSun, 15 Dec 2013 03:17:36 +0100https://ask.sagemath.org/question/10574/why Z5 in Qp different with Zp(5)https://ask.sagemath.org/question/10832/why-z5-in-qp-different-with-zp5/K = Qp(5, print_mode='digits')
Z5 = K.integer_ring();Z5;Z5;Z5 ==Zp(5)
5-adic Ring with capped relative precision 20
5-adic Ring with capped relative precision 20
FalsecjshSun, 15 Dec 2013 03:12:15 +0100https://ask.sagemath.org/question/10832/Maximal unramified extension of Qphttps://ask.sagemath.org/question/7970/maximal-unramified-extension-of-qp/Hello,
is it possible to deal with the maximal unramified extension of Qp in Sage?
If so, then please post a code example! Thanks!
Bye
Larslars.tennstedtMon, 11 Nov 2013 10:46:00 +0100https://ask.sagemath.org/question/7970/Roots of p-adic polynomialshttps://ask.sagemath.org/question/10695/roots-of-p-adic-polynomials/Hello again,
I want to get the roots of a p-adic polynomial. I wrote the following code:
p = 2
q = 4
K = Qp(p)
L.<omega> = Qq(q)
O_L = L. integer_ring()
pi = L.uniformizer()
q = L.residue_class_degree()
f = X^q + pi*X
f.roots()
This rises following error:
NotImplementedError: root finding for this polynomial not implemented
I still use version 5.10 of Sage. The compile of the latest version 5.12 is ongoing. Do I miss something or is there no way to get the roots in this case?
Will this be implemented in the future or does someone know a workaround?
Thanks for your time again!
Bye
Lars
lars.tennstedtMon, 04 Nov 2013 14:54:57 +0100https://ask.sagemath.org/question/10695/p-adic power serieshttps://ask.sagemath.org/question/10688/p-adic-power-series/Hello,
I am new to Sage and I want to to do some math on p-adic power series. I want to
define such a power series but I did not succeeded. The last line of the
following code rises an exception:
p = 2
q = 4
K = Qp(p)
L.<omega> = Qq(q)
O_L = L.integer_ring()
R.<X> = PowerSeriesRing(O_L)
pi = L.uniformizer()
q = L.residue_class_degree()
f(X) = X^q + pi*X
**TypeError: unsupported operand parent(s) for '*': '2-adic Field with capped
relative precision 20' and 'Symbolic Ring'**
Can someone help me, please? Thanks for your time!
Bye
Larslars.tennstedtSun, 03 Nov 2013 05:23:41 +0100https://ask.sagemath.org/question/10688/Artin decomposition for p-adic numbershttps://ask.sagemath.org/question/10663/artin-decomposition-for-p-adic-numbers/How can I decompose a p-adic number
... d_2 d_1 d_0. d_{-1} ... d_{-k}
into its integer part
d_2 d_1 d_0.
and fractional part
. d_{-1} ... d_{-k} ?
The does not seem to exist a kind of floor function.
Klaus ScheicherSun, 27 Oct 2013 03:06:49 +0100https://ask.sagemath.org/question/10663/what PI and e in p-adic number?https://ask.sagemath.org/question/10631/what-pi-and-e-in-p-adic-number/what PI and e in p-adic number?cjshSun, 20 Oct 2013 04:04:09 +0200https://ask.sagemath.org/question/10631/is there a field name Cp(above bar) in p-adic ?https://ask.sagemath.org/question/10630/is-there-a-field-name-cpabove-bar-in-p-adic/is there a field name Cp(above bar) in p-adic ?cjshSun, 20 Oct 2013 04:00:20 +0200https://ask.sagemath.org/question/10630/Solving polynomial equations over p-adic fieldshttps://ask.sagemath.org/question/9860/solving-polynomial-equations-over-p-adic-fields/Hi - as usual I fear my naiv-IT is coming to the fore here, but I have been going around in circles on this for 3 days and I need help please!!
There are 2 basic questions arising from the same thing:
(1) Is "solve" supposed to be implemented for p-adic numbers at all? I can get solutions to things in finite fields but when I try to "lift" them using O(p^n) etc it all goes wrong.
(2) Quite apart from that, why can I not use "solve" using the "variables" (for which I want solutions) as the indeterminates in a polynomial ring over which the equations are already defined? For example, if I define my polynomial ring via:
sage: R.<X> = Zq(3^4,2);
sage: RAB.< a,b> = R[];
and if I then try
sage: solve([a+b==6,a-b==2],[a,b])
it tells me that "a is not a valid variable".GaryMakWed, 27 Feb 2013 18:56:26 +0100https://ask.sagemath.org/question/9860/p-adic field as a vector space?https://ask.sagemath.org/question/10302/p-adic-field-as-a-vector-space/Hello!
Is there an equivalent of absolute(or relative)_vector_space for p-adic fields?
I couldn't find anything! If not, is there any other way to realize a p-adic field extension as a vector space (over Q, but even over Q_p)?
I'm trying to use linear_dependence in particular.
Actually my main goal is to do something like what algdep does, but that command only seem to work for Q_p and not any extension!
I apologize for being an absolute illiterate with computers which shall show itself sooner or later!bamirSun, 30 Jun 2013 23:27:43 +0200https://ask.sagemath.org/question/10302/extracting digits in p-adic expansionhttps://ask.sagemath.org/question/9335/extracting-digits-in-p-adic-expansion/Here is a maybe naive question.
I am working over an unramified extansion of $\mathbb{Z}_p$ (let say of degree 2), say R.(c) = Zq(7^2) so that every element of R is written as ($a_0$ *c + $b_0$) + ($a_1$ *c + $b_1$) *7 + ($a_2$*c + $b_2$)*7^2 + $\ldots$ .
Is there a command to extract each of the $a_i$ or $b_i$ individually ?A MTue, 15 Jan 2013 10:20:05 +0100https://ask.sagemath.org/question/9335/