Wikipedia:Reference desk/Archives/Mathematics/2009 August 28

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August 28

Automorphisms

I want to prove the following result: Suppose I have an irreducible factor of the polynomial $x^{n}-1$, say h(x) in $\mathbb {Z} _{p}[x]$ (p prime). Now my first question is: If it is said that the Frobenius automorphism fixes h(x) does it mean that the coefficients of h(x) are permuted. Why does it happen? Secondly supposing we consider a splitting field in which the roots of h(x) exist. Will the Frobenius automorphism (over the splitting field Z_p(a,b,c...)) permute the roots a,b,c... and why? Finally why can't we define a splitting field for a polynomial defined over any general ring as opposed to over fields as the article Splitting field says? Thanks--Shahab (talk) 05:21, 28 August 2009 (UTC)

(I may be misunderstanding your questions, so if you do not find my answers helpful please ask again.) The Frobenius automorphism $\sigma$ on $K=\mathbb {Z} _{p}$ is the identity (there are no non-identity automorphisms of $\mathbb {Z} _{p}$). Therefore $\sigma$ fixes all polynomials over K, including h(x), no matter what h(x) is. Now let L be the splitting field of h over K, and let $\sigma$ be the Frobenius automorphism of L. Since $\sigma$ fixes K (why do we know this?), $\sigma h=h$, so the roots of h and $\sigma h$ in L are the same, and thus $\sigma$ permutes the roots of h; that is, $\sigma$ is in the Galois group of L over K.

Your last question is somewhat subtler. Recall that when we wish to adjoin a root of an irreducible polynomial f to a field K, we construct the field $L=K[X]/(f(X))$. (This is a field because (f(X)) is a prime ideal in the ring K[X], which is true because f is irreducible.) There is nothing really stopping us from doing the same operation to a ring R; fundamentally, the reason why we don't do this is because the results are uninteresting (I haven't verified this myself, but I assume that that is so, otherwise we would talk about splitting rings). However if we try to construct these "splitting rings" we can sometimes lose nice properties. For example, let $R=\mathbb {Z} /4\mathbb {Z}$, the cyclic ring of 4 elements, and let $f(x)=2x+1$. Let $S=R[X]/(f(X))$; then S has only one element, so there is no embedding of R into S anymore. Eric. 98.207.86.2 (talk) 07:48, 28 August 2009 (UTC)

Thanks Eric. I understand that the Frobeinus map over $\mathbb {Z} _{p}$ is the identity due to Fermat's little theorem. My main problem was the second question which is still not clear as it was incompletely asked. What I really want to do is this: Given that $g(x)\in \mathbb {Z} _{p^{r}}[x]$ is such that $g\equiv h(\mod p)$ (which I take to mean that the coefficients of g(x) and h(x) agree mod p) and g(x) is the irreducible factor of $x^{n}-1$ then the Frobenius map permutes the roots of g(x). Now $\mathbb {Z} _{p^{r}}$ is the ring in question and the roots must lie in the splitting field hence my last question. Assuming I have some splitting field F of g(x) I reason like this: (All computations are mod p and $\sigma$ is the Frobenius automorphism) g=h and so $\sigma (g)=\sigma (h)$ (where $\sigma$ is really acting upon the coefficients). But $\sigma (h)=h$ and so $\sigma (g)=h=g$. Now I want to use the fact(?) that automorphisms permute the roots of irreducible polynomials when the coefficients are fixed by the automorphism (I'll apply the Frobenius map on F). My second question is that I want a proof of this. Also please excuse me if some of what I have written is obvious or redundant. Since I am mostly self taught so there are quite a few gaps in my knowledge. Thanks again--Shahab (talk) 09:17, 28 August 2009 (UTC)

Are you trying to understand how to lift polynomial factorizations from Z/pZ to the p-adic integers, as in the algorithm to factor integer polynomials described in the original LLL paper? You can probably do that more easily than by developing a Galois theory for arbitrary commutative rings. JackSchmidt (talk) 14:41, 28 August 2009 (UTC)

Unfortunately I'm confused. I see that are you given an irreducible polynomial g (by "the irreducible factor" of $X^{n}-1\in \mathbb {Z} [X]$ I suppose you mean the one with degree $\phi (n)$, which you then project into $\mathbb {Z} _{p^{r}}[X]$ by taking the coefficients mod $p^{r}$) over $\mathbb {Z} _{p^{r}}$. However you introduce two new polynomials h and f and I don't know what those are. You also mention the Frobenius map on $\mathbb {Z} _{p^{r}}$; however, in general, this map (of raising to the power p) is not a ring homomorphism when $r>1$, so it would be unlikely to permute the roots of g or do anything nice.

Your last fact, that field automorphisms permute the roots of irreducible polynomials when the coefficients are fixed by the automorphism is correct. Eric. 216.27.191.178 (talk) 20:09, 28 August 2009 (UTC)

I made the mistake of writing f when I really meant h. I'll try to explain myself more clearly now. (BTW Theorem 1 of this paper online is the closest thing I could find on the internet to what I am trying to prove). I have some fixed irreducible factor of x^n-1 in Z_p[x] which is h. In Z_p^r[x] the same h goes by the name g. The roots of g lie in some extension field F (I don't know how such a field can be constructed. Perhaps we take the quotient field of Z_p^r and then find the splitting field) of Z_p^r. I want to raise all elements of F to pth power and then show that roots are only permuted. Thanks--Shahab (talk) 08:02, 29 August 2009 (UTC)

How do I prove the fact that that field automorphisms permute the roots of irreducible polynomials when the coefficients are fixed by the automorphism is correct.--Shahab (talk) 08:02, 29 August 2009 (UTC)

(dedent) Ah, that is clear. First I'll address the easy part (namely, the part I know how to prove). Suppose $\sigma$ is an automorphism of the field L, the fixed field of $\sigma$ is K, and f is a polynomial over K that splits completely in L; we wish to prove that $\sigma$ permutes the roots of f. For any polynomial g over L and $\alpha \in L$, we have $\sigma (g(\alpha ))=(\sigma g)(\sigma \alpha )$ (because $\sigma$ is an automorphism of L), so $\sigma$ maps the roots of g bijectively to the roots of $\sigma g$. In the case of f, we know $f=\sigma f$ (since the coefficients of f are fixed by $\sigma$); thus $\sigma$ bijectively maps the roots of f to the roots of f, i.e., permutes the roots of f.

As for what you're trying to prove, we run into difficulties. (The ring $R=\mathbb {Z} _{p^{r}}$ does not have a quotient field for r > 1 because R is not an integral domain; in particular $p\cdot p^{r-1}=0$.) I ran through an example to see if I could find the result you were looking for: take p = 2, r = 2, and $f(x)=x^{2}-x+1\in \mathbb {Z} _{p}[x],g(x)=x^{2}-x+1\in R[x]$. f and g are irreducible, and divide $x^{6}-1$. If we square the coefficients of g, we get another polynomial $x^{2}+x+1\in R[x]$, which being unequal to g, cannot have the same roots (even were we working in a ring where they both had all their roots). Eric. 98.207.86.2 (talk) 11:32, 29 August 2009 (UTC)

The best bet for understanding this is to just work with the p-adics. Use Hensel's lift to get the factorization over the p-adic integers, find the ring of integers A of the splitting field of the p-adic polynomial, the galois group is abelian so you can take an arbitrary factor of (p) as P and then look at A/P^r as the "extension field" of Z/p^rZ. Make sure that n and p are coprime (6 and 2 is naughty) otherwise the factorization can change. The paper you mentioned has a nice reference [2] that explains the LLL Z[x] factorization algorithm. The permutation of the roots happens in A/P^r the same as it does in A and the same as it does in A/P. If you are naughty and take n and p to have common factors, then the number of roots is not stable, and so the permutation actions cannot possibly be equivalent. JackSchmidt (talk) 13:06, 29 August 2009 (UTC)

That sounds quite reasonable to me. I am wondering if you could explain a few details for me. We have $L=\mathbb {Q} _{p}(\zeta _{n})$ is the splitting field of $x^{n}-1$ over $K=\mathbb {Q} _{p}$. I forget why L/K is abelian (cyclic, right?) but I can look that up myself. If n and p are coprime then L/K is unramified, but I don't see what that gets us. If f is a polynomial over $R=\mathbb {Z} _{p}$ we can project it to $R/p=\mathbb {Z} /p$, and that projection should split completely in A/P, where we have a Frobenius automorphism that permutes its roots in A/P, However I don't see how that helps us in A and A/P^r. Also, what do you mean by "the number of roots is not stable"? Eric. 98.207.86.2 (talk) 22:31, 29 August 2009 (UTC)

The Galois action on the field of fractions of A, L, restricts to a nice action on A itself. That action permutes the n roots of x^n-1 in A, and that action is permutation isomorphic to the action of the Galois group of A/P over Z/pZ on the roots of x^n-1. However, if n and p are not coprime, then there is a different number of n'th roots of unity in L versus A/P, so the actions cannot be permutation isomorphic. For example, if n=2 and p=2, then one has that both L=Q_2 and A=Z_2 have 2 square roots of unity, but Z/2Z=A/P has only one, Z/4Z has two, and Z/8Z has four. JackSchmidt (talk) 00:35, 30 August 2009 (UTC)

Thanks. I see my trouble now, I had gotten myself confused about the definition of Frobenius map#Frobenius_for_local_fields. Eric. 98.207.86.2 (talk) 01:35, 30 August 2009 (UTC)

Nonconvex polyhedra

I have a question: why is it the polyhedra {5/2, 3} and {5/2, 5} both exist but {5/2, 4} doesn't? Professor M. Fiendish, Esq. 12:38, 28 August 2009 (UTC)

The article on the four Kepler-Poinsot polyhedra states that "Augustin Cauchy proved the list complete by stellating the Platonic solids, and almost half a century after that, in 1858, Bertrand provided a more elegant proof by facetting them." Therefore, I suppose that from the definition of a regular star-polyhedron it follows directly that its vertices span a Platonic polyhedron as convex hull. If this supposition is correct, making a complete list is just a matter of checking a finite number of cases. (I wrote "I suppose" because I did not look for the exact definition. Of course, one can give apparently weaker definitions, but equivalent a posteriori, that make the proof more difficult and more general.) --pma (talk) 18:05, 28 August 2009 (UTC)

{5/2, 4} would have axes of 5-fold and 4-fold rotational symmetry; but these can coexist only in a hyperbolic geometry. —Tamfang (talk) 23:55, 9 September 2009 (UTC)

nested gaussian integrals.

First off: This is not for a homework problem

I am doing some work which requires calculation of nested time integrals (which come up in perturbation theory). I was able to work out a nice formula for a second order equation of the form

${\begin{aligned}&\int _{-\infty }^{\infty }dt_{2}\int _{-\infty }^{t_{2}}dt_{1}\exp \left(-{\frac {t_{2}^{2}}{2\sigma _{2}^{2}}}-i\Delta _{2}t_{2}\right)\exp \left(-{\frac {t_{1}^{2}}{2\sigma _{1}^{2}}}-i\Delta _{1}t_{1}\right)\\&\ \ \ \ \ \ =\pi \sigma _{1}\sigma _{2}\exp \left(-{\frac {1}{2}}(\Delta _{1}^{2}\sigma _{1}^{2}+\Delta _{2}^{2}\sigma _{2}^{2})\right)\operatorname {erfc} \left(i{\frac {\Delta _{2}\sigma _{2}^{2}-\Delta _{1}\sigma _{1}^{2}}{\sqrt {2\sigma _{1}^{2}+2\sigma _{2}^{2}}}}\right)\end{aligned}}$ .

Since there are nice efficient ways to calculate the error function, I don't need to do any time-consuming numerical integration for the second-order nested integral. But I am stuck coming up with a similar solution to a third-order nested integral.

$\int _{-\infty }^{\infty }dt_{3}\int _{-\infty }^{t_{3}}dt_{2}\int _{-\infty }^{t_{2}}dt_{1}\exp \left(-{\frac {t_{3}^{2}}{2\sigma _{3}^{2}}}-i\Delta _{3}t_{3}\right)\exp \left(-{\frac {t_{2}^{2}}{2\sigma _{2}^{2}}}-i\Delta _{2}t_{2}\right)\exp \left(-{\frac {t_{1}^{2}}{2\sigma _{1}^{2}}}-i\Delta _{1}t_{1}\right)$

Any help would be greatly appreciated. mislih 20:32, 28 August 2009 (UTC)

I think you can express the triple integral in terms of the erf function [or antiderivatives of it] (and, possibly, even the analog multiple integral). I'll give you some hints here below.

• You want an integral of a function of the form exp(-Ax²-By²-Cz²-2ax-2by-2cz) on the set {x<y<z}. Here A,B,C are positive constant, and a,b,c are actually imaginary numbers. Notice that {x<y<z} is the intersection of two half-spaces, with an angle of π/3.
• Since the integral is clearly an entire function of a,b,c, you can do the computation just for all real a,b,c and then extend by analytic continuation the result to all complex a,b,c; the extension will be automatic (recall that erf(z) is an entire function).
• Now complete the square, that is, here, use the identity exp(-(u+k)²)=exp(-u²-2ku)exp(-k²), and then change variable. This way you transform the initial into the integral of exp(-x²-y²-z²) on a suitable domain which is the intersection of two affine half-spaces, whose angle and position wrto the origin, of course, depend on all the initial coefficients (there is also a factor coming from the complete-the-square thing and from the Jacobian of the affine transformation of the variables).
• Thanks to the rotation invariance of exp(-x²-y²-z²), you can rotate the domain of integration so that its edge (i.e. the singular line of the boundary) is a line parallel to the z axis. A partial integration in the z variable gives you a factor sqrt(π) in front of the double integral of exp(-x²-y²) on an angular domain on the x,y plane (that is, the orthogonal section of the previous three-dimensional domain).
• Again by rotation invariance, you may assume that the angular domain is bounded by a half-line parallel to the y-axis, and you are left with the integral of exp(-x²-y²) on this domain. A first integration produces a partial integral of the form erf(αx+β)exp(-x²), that you still have to integrate in x>0. (I don't see yet a formula for the latter antiderivative of erf(αx+β)exp(-x²); but if the matter is just about numerics, it certainly has a very fastly convergent power series expansion).

I hope it's clear enough; of course to complete the computation you have some work to do in order to write everything in terms of the initial data. In case ask here again and we'll try to answer.--pma (talk) 21:47, 29 August 2009 (UTC)