Eisenstein polynomial of (relative) extensions

[saraedum]

Someone asked per email:

[…] For example, in the following code

from henselization import *
from henselization.benchmarks.splitting_fields import splitting_field
K = QQ.henselization(3)
R.<T> = K[]
f=T^9-3
splitting_field(f)
Extension defined by a54^6 + a9^3*a54^3 + a9^6 of Extension defined by a9^9 - 3 of Henselization of Rational Field with respect to 3-adic valuation

It defines the relative extension […] which is not Eisenstein over [the intermediate field]. This is reasonable but somehow inconvenient. Someone told me that the corresponding Eisenstein polynomial can be achieved by using Ore-MacLane/ Okutsu-Montes algorithm, but I can't see their connection. The only software supporting such feature may be MAGMA (MAGMA calculates the Eisenstein polynomial of the (totally ramified) splitting field of polynomials with Z coefficients in Q_p, which is not open-sourced. So I wonder if you have any idea about this problem?

[saraedum]

If you only need the Eisenstein polynomial over Q_p, you can do:

sage: K.base().polynomial()
e54^54 - 9*e54^48 - 9*e54^45 + 9*e54^39 - 3*e54^36 + 27*e54^28 + 27*e54^27 + 54*e54^25 + 27*e54^24 + 54*e54^22 + 27*e54^19 + 36*e54^18 + 27*e54^16 + 54*e54^15 + 27*e54^13 + 18*e54^12 + 54*e54^4 + 54*e54^3 + 3

That's defining the extension of Q that Henselization is using internally.

We can verify that the polynomial actually splits in linear factors over this field:

sage: R.<e54> = Qp(3)[]
sage: L.<e54> = Qp(3).extension(K.base().polynomial().change_ring(Qp(3)))
sage: v.montes_factorization(T^9 - 3, assume_squarefree=True, required_precision=1)
((1 + O(e54^1080))*T + e54^6 + e54^10 + 2*e54^14 + 2*e54^15 + 2*e54^18 + e54^19 + e54^21 + 2*e54^22 + e54^27 + 2*e54^28 + e54^29 + e54^30 + 2*e54^31 + e54^32 + 2*e54^33 + e54^36 + e54^37 + e54^38 + e54^39 + 2*e54^40 + e54^41 + 2*e54^42 + 2*e54^43 + e54^44 + 2*e54^45 + 2*e54^46 + e54^47 + e54^49 + 2*e54^53 + O(e54^1086)) * ((1 + O(e54^1080))*T + e54^6 + e54^10 + 2*e54^14 + 2*e54^15 + 2*e54^18 + e54^19 + e54^21 + 2*e54^22 + e54^27 + 2*e54^28 + e54^29 + e54^30 + 2*e54^31 + e54^32 + e54^33 + e54^36 + e54^38 + e54^39 + 2*e54^40 + 2*e54^41 + 2*e54^43 + e54^44 + e54^46 + e54^47 + 2*e54^48 + 2*e54^49 + 2*e54^53 + O(e54^1086)) * ((1 + O(e54^1080))*T + e54^6 + e54^10 + 2*e54^14 + 2*e54^15 + 2*e54^18 + e54^19 + e54^21 + 2*e54^22 + e54^27 + 2*e54^28 + e54^29 + e54^30 + 2*e54^31 + e54^32 + e54^36 + 2*e54^37 + e54^38 + e54^39 + 2*e54^40 + e54^42 + 2*e54^43 + e54^44 + e54^45 + e54^47 + e54^48 + 2*e54^53 + O(e54^1086)) * ((1 + O(e54^1080))*T + e54^6 + e54^10 + 2*e54^14 + 2*e54^18 + 2*e54^19 + e54^21 + 2*e54^22 + 2*e54^23 + e54^24 + 2*e54^28 + e54^29 + 2*e54^30 + e54^31 + 2*e54^32 + 2*e54^33 + e54^37 + 2*e54^38 + e54^39 + 2*e54^40 + 2*e54^42 + 2*e54^43 + e54^44 + 2*e54^45 + e54^47 + 2*e54^48 + e54^49 + e54^51 + 2*e54^52 + e54^53 + O(e54^1086)) * ((1 + O(e54^1080))*T + e54^6 + e54^10 + 2*e54^14 + 2*e54^18 + 2*e54^19 + e54^21 + 2*e54^22 + 2*e54^23 + e54^24 + 2*e54^28 + e54^29 + 2*e54^30 + e54^31 + 2*e54^32 + 2*e54^37 + 2*e54^38 + e54^39 + 2*e54^40 + 2*e54^41 + 2*e54^42 + 2*e54^43 + e54^44 + e54^45 + 2*e54^46 + e54^47 + 2*e54^50 + 2*e54^51 + 2*e54^52 + e54^53 + O(e54^1086)) * ((1 + O(e54^1080))*T + e54^6 + e54^10 + 2*e54^14 + 2*e54^18 + 2*e54^19 + e54^21 + 2*e54^22 + 2*e54^23 + e54^24 + 2*e54^28 + e54^29 + 2*e54^30 + e54^31 + 2*e54^32 + e54^33 + 2*e54^38 + e54^39 + 2*e54^40 + e54^41 + 2*e54^42 + 2*e54^43 + e54^44 + e54^46 + e54^47 + e54^48 + 2*e54^49 + e54^50 + 2*e54^52 + e54^53 + O(e54^1086)) * ((1 + O(e54^1080))*T + e54^6 + e54^10 + 2*e54^14 + e54^15 + 2*e54^18 + e54^21 + 2*e54^22 + e54^23 + 2*e54^27 + e54^29 + 2*e54^32 + 2*e54^33 + e54^36 + 2*e54^39 + e54^40 + e54^41 + 2*e54^43 + e54^44 + 2*e54^45 + 2*e54^47 + 2*e54^49 + 2*e54^50 + e54^51 + e54^52 + O(e54^1086)) * ((1 + O(e54^1080))*T + e54^6 + e54^10 + 2*e54^14 + e54^15 + 2*e54^18 + e54^21 + 2*e54^22 + e54^23 + 2*e54^27 + e54^29 + 2*e54^32 + e54^33 + e54^36 + 2*e54^37 + 2*e54^39 + e54^40 + 2*e54^41 + 2*e54^42 + 2*e54^43 + e54^44 + 2*e54^47 + 2*e54^48 + e54^50 + e54^51 + e54^52 + O(e54^1086)) * ((1 + O(e54^1080))*T + e54^6 + e54^10 + 2*e54^14 + e54^15 + 2*e54^18 + e54^21 + 2*e54^22 + e54^23 + 2*e54^27 + e54^29 + 2*e54^32 + e54^36 + e54^37 + 2*e54^39 + e54^40 + e54^42 + 2*e54^43 + e54^44 + e54^45 + 2*e54^47 + e54^48 + e54^49 + e54^51 + e54^52 + O(e54^1086))
sage: len(_)
9

[saraedum]

If you want to understand how this works, it's implemented here:

henselization/henselization/sage/rings/padics/henselization/henselization.py

Line 1444 in bded9ba

def _eisenstein_model(self, base, name=None):

The idea (as far as I recall) is to take a uniformizing element (which is can be constructed from the MacLane description of the valuation, see

henselization/henselization/sage/rings/padics/henselization/henselization.py

Line 557 in bded9ba

def uniformizer(self):

and then you'll have to follow the implementation into the SageMath code in sage.rings.valuation.) We then compute the charpoly of this element and hope that it's irreducible. If it's not, we perturb the uniformizing element a bit to get another uniformizing element. Repeating this process we have to eventually find an irreducible charpoly because of the primitive element theorem.

[saraedum]

If you wanted to only write the relative extension with an Eisenstein polynomial, the same approach should work. However, the Henselization code cannot compute the matrix of an element w.r.t. a relative extension yet, it currently fails because of this:

henselization/henselization/sage/rings/padics/henselization/henselization.py

Line 1074 in bded9ba

raise NotImplementedError("Basis with respect to %s"%base)

Even with that fixed, I am not sure that the arithmetic of the elements of the Henselization would allow us to compute a charpoly directly.

You can, however, also use the valuation code in SageMath directly to compute that polynomial:

sage: R.<a9> = QQ[]
sage: K.<a9> = QQ.extension(a9^9 - 3)
sage: R.<a54> = K[]
sage: L.<a54> = K.extension(a54^6 + a9^3*a54^3 + a9^6)
sage: v = L.valuation(3)
sage: f = v.uniformizer().matrix(base=K).charpoly()
sage: [v(c) for c in f.coefficients()]
[1/9, 1/3, 2/9, 2/9, 1/9, 7/9, 0]

[YijunYuan]

If you want to understand how this works, it's implemented here:

henselization/henselization/sage/rings/padics/henselization/henselization.py

Line 1444 in bded9ba

def _eisenstein_model(self, base, name=None):

The idea (as far as I recall) is to take a uniformizing element (which is can be constructed from the MacLane description of the valuation, see

henselization/henselization/sage/rings/padics/henselization/henselization.py

Line 557 in bded9ba

def uniformizer(self):

and then you'll have to follow the implementation into the SageMath code in sage.rings.valuation.) We then compute the charpoly of this element and hope that it's irreducible. If it's not, we perturb the uniformizing element a bit to get another uniformizing element. Repeating this process we have to eventually find an irreducible charpoly because of the primitive element theorem.

Is there any self-contained reference that explains the mechanism and theory behind the Henselization package? I'm lost in jumping through the code. In detail, I don't understand how the uniformizer is given by the limit valuation, and what's the matrix of it?

[saraedum]

Is there any self-contained reference that explains the mechanism and theory behind the Henselization package?

There is no proper documentation for this package unfortunately. The underlying MacLane valuation theory is explained in Chapter 4 of my thesis.

I'm lost in jumping through the code.

Sorry for that. I never had the time to write something up to specifically explain how this code works.

In detail, I don't understand how the uniformizer is given by the limit valuation

Given an extension given as , suppose the valuation on L is given by a limit valuation which has . To get a uniformizer find some sum of the that sums to 1/n. Multiplying the same gives you a uniformizer.

and what's the matrix of it?

Sorry, I meant the matrix of multiplication of that element. (Given an extension L/K, consider L as a K vector space and fix a basis . Given an element x in L, write the in the basis B and then just organize this in a matrix.) The implementation of this is very short but might be cryptic:

henselization/henselization/sage/rings/padics/henselization/base_element.py

Lines 293 to 294 in c92ae9d

	V,f,g = self.parent().module(base=base)
	return V.hom([g(f(b)*self) for b in V.basis()]).matrix()

[YijunYuan]

I spend some time reading Section 4 of your thesis, where you mainly focus on the function field (am I right?), instead of the $p$-adic case (the theory also works in the mixed-characteristic setting, just like the Henselization package does, right?). The key to constructing such an Eisenstein polynomial is to find a uniformizer of the extension (basically, this is equivalent to characterizing the valuation on the extension) and the following matrix thing is easy to understand after I review the code. I'm still reading Section 4.6 of your paper intensively. Maybe some further questions will appear.:neckbeard:

[saraedum]

Yes, I wrote Section 4 with function fields in mind so I could apply the theory to describe models of curves. But it works the same for the p-adic case.

[YijunYuan]

Let $K=\mathbb{Q}_3(\zeta_3)$, with uniformizer of its valuation $v_K$ given by $\zeta_3-1$. Let $G=t^3-3$. I want to extend $v_K$ to $K[t]/(G)$, i.e. finding a discrete pseudo-valuation $v$ on $K[t]$, which extends the Gauss valuation $v_0$ induced by $v_K$ and sends exactly $(G)$ to infinity. Since $K$ is complete, by Corollary 4.67 $v$ is an infinite inductive valuation. (right?) The following is an imitation of your Example 4.78:

1. We start with $v_0$, which has residue field $\mathbb{F}_3(t)$. To find out the key polynomial over $v_0$, we consider the reduction of $G$ in $\mathbb{F}_3(t)$: $\bar{G}=t^3$. Therefore we may take $\phi_1=t$, which makes $v_1=[v_0,v_1(\phi_1)=\lambda_1]$ into an approximation of $v$. Now we have to find out the value of $\lambda_1$. The $\phi_1$-adic expansion of $G$ is given by $G=\phi_1^3-3$. By Lemma 4.77, $3\lambda_1$ should equals to $2$, i.e. $\lambda_1=\frac{2}{3}$.
2. To go on, we construct the key polynomial over $v_1$. Using the formula that appears at the top of page 26, the reduction of $G$ in the residue ring of $v_1$ is given by $H_0(a_0)y_1^0+H_0(a_3\cdot (\zeta_3-1)^2)y_1^1=0$. I don't think it makes sense...

What's wrong here?

[saraedum]

By Lemma 4.77, $3\lambda_1$ should equals to $2$, i.e. $\lambda_1=\frac{2}{3}$.

You get Since you get . If I understood your example correctly, you somehow got a 2 in your computation that looks wrong to me.

To go on, we construct the key polynomial over $v_1$.

The reduction of G is zero. That looks correct to me. To construct the key polynomial via reduction you need to shift G to valuation zero first. You should find that the reduction is something trivial like y₁ which means that your next key polynomial will have degree . With these degrees, nothing else could have happened here: G has degree 3 and you already found a ramification of degree 3 with λ₁, so G must be a key polynomial now.

Here's the computation with SageMath:

sage: K.<ζ> = NumberField(x^2 + x + 1)
sage: vK = K.valuation(3)
sage: vK.uniformizer()
ζ + 2
sage: R.<t> = K[]
sage: G = t ^ 3 - 3
sage: L.<t> = K.extension(G)
sage: vK.extensions(L)
[3-adic valuation]
sage: v = _[0]
sage: w = v._base_valuation
sage: w
[ Gauss valuation induced by 3-adic valuation, v(t) = 1/3 , … ]
sage: w._approximation
[ Gauss valuation induced by 3-adic valuation, v(t) = 1/3 ]
sage: w._improve_approximation()
sage: w._approximation
[ Gauss valuation induced by 3-adic valuation, v(t) = 1/3, v(t^3 - 3) = +Infinity ]

[YijunYuan]

You get Since  = v(t^3)=\lambda_1) you get . If I understood your example correctly, you somehow got a 2 in your computation that looks wrong to me.

Well it seems that I normalized the valuation on $K$ to $v_K(\zeta_3-1)=1$, i.e. $v_K(3)=2$ which leads to the problem. But why it's not acceptable? Instead of viewing $K$ as an extension of $\mathbb{Q}_3$ and using the $3$-adic valuation, using the normalized valuation on $K$ seems to be a natural choice. As you wrote in the Conventions and Standing Assumptions part of your article,

If we do not discuss extensions of valued fields, then all valuations are normalized such that a uniformizer $\pi\in K$ has valuation $v_K(\pi)=1$.

[saraedum]

You can normalize like that. It should not make a difference here.

[YijunYuan]

Let be an approximation of pseudo-valuation over (K,v) which sends G to infinity. Now we need to construct the next key polynomial . This can be divided into following steps, if I understand the algorithm correctly:

1. Shift G to valuation 0. At the first glance, this can be easily done by multiplying a polynomial with , but does S impact the divisible relationship? More specifically, since G is -divisible by , the shifting operation turns it into the condition that SG is -divisible by (G may not have -reciprocal, so this relationship is not equivalent to ). Anyway, we set .
2. Factorize in the residue field of . This can be done by using the formula in Section 4.1.3 of your thesis. We choose an irreducible factor of it. The second doubt: if K is complete and G is irreducible over K, the choice of the irreducible factor has no impact(up to equivalence) on the final outcome, i.e. the infinite inductive pseudo-valuation, right? The extension should be unique.
3. Lift to a key polynomial. This is not documented in your thesis. From the source code of SageMath, it's based on the proof of Theorem 13.1 of [Mac1936I], which uses different notations with your thesis. I have difficulty understanding this, so could you give me some hints? The first step is clear: lift back to K[t]. We denote the lifting by . Now what to do next?

[saraedum]

To 1. You are right, the S needs to be chosen appropriately otherwise divisibility changes. I believe that lemmata in 4.1.2 are talking about that essentially. Choosing an S as constructed in these lemmata should work but I forget the details now.

To 2. Exactly, that choice should not matter.

To 3. What to do next: I believe that you need to multiply by some S to shift the valuation to what it needs to be to be a key polynomial and then fix the leading coefficient (because key polynomials are assumed to be monic.) Does that make sense? I can try to figure out how this works exactly. But I don't remember the details right now.

[YijunYuan]

See here for the math-rended version.

To 3. What to do next: I believe that you need to multiply by some S to shift the valuation to what it needs to be to be a key polynomial and then fix the leading coefficient (because key polynomials are assumed to be monic.)

Here is my thought: Our goal is to find $\phi_{n+1}$ which $v_n$-divides $G$. By multiplying an $v_n$-unit with proper valuation on $G$, we may assume that $v_n(G)=0$. We factorize $G$ in $k_{v_n}$ and get a factor $\bar{\phi}_{n+1}$. By Theorem 13.1 of [Mac1936I], there exists a key polynomial $\phi_{n+1}$ satisfying that the reduction of $\phi_n^{\tau_n\flat_n\text{deg}\ \bar{\phi}_{n+1}}\phi_{n+1}$ to $k_{v_n}$ is $\bar{\phi}_{n+1}$. Here $\phi_n^{\tau_n\flat_n\text{deg}\ \bar{\phi}_{n+1}}$ is a $v_n$-unit (right?) to shift the valuation. Until now, everything is right?

As we have discussed, the first step of the construction in Theorem 13.1 of [Mac1936I] is lifting $\bar{\phi}_{n+1}$ to $\tilde{\phi}_{n+1}\in K[t]$. After that,

multiply $\tilde{\phi}_{n+1}$ by $\phi_{n}^{\tau_n\sharp_n \text{deg}\ \bar{\phi}_{n+1}}$ and in the expansion of the resulting product drop all terms not of minimum value and then replace the leading coefficient of $\phi_n$ by $1$.

I'm not sure what's happening here. The term $\phi_{n}^{\tau_n\sharp_n \text{deg}\ \bar{\phi}_{n+1}}$ seems to be used to shift the valuation to $0$? The following steps are also confusing, especially

replace the leading coefficient of $\phi_n$ by $1$.

Of course, it's used to make the polynomial monic. But why should I replace the leading coefficient of a key polynomial, instead of the expansion of the product?

P.S. Sorry for the poor math reading experience. I'm using a chrome extension (I modified it a little) to render MathJax. Next time I will try to use Github's math render, as you do.
P.P.S. Thanks for your patience. It's really helpful.