Splitting Behavior of SnS_n-Polynomials

We analyze the probability that, for a fixed finite set of primes S, a random, monic, degree n polynomial f(x) with integer coefficients in a box of side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with splitting field over the rationals having Galois group $S_n$; (ii) the polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii) f(x) has a prescribed splitting type at each prime p in S. The limit probabilities as $B \to \infty$ are described in terms of values of a one-parameter family of measures on $S_n$, called splitting measures, with parameter $z$ evaluated at the primes p in S. We study properties of these measures. We deduce that there exist degree n extensions of the rationals with Galois closure having Galois group $S_n$ with a given finite set of primes S having given Artin symbols, with some restrictions on allowed Artin symbols for p<n. We compare the distributions of these measures with distributions formulated by Bhargava for splitting probabilities for a fixed prime $p$ in such degree $n$ extensions ordered by size of discriminant, conditioned to be relatively prime to $p$.Comment: 33 pages, v2 34 pages, introduction revise

The emergence of 4-cycles in polynomial maps over the extended integers

Let $f(x) \in \mathbb{Z}[x]$; for each integer $\alpha$ it is interesting to consider the number of iterates $n_{\alpha}$, if possible, needed to satisfy $f^{n_{\alpha}}(\alpha) = \alpha$. The sets $\{\alpha, f(\alpha), \ldots, f^{n_{\alpha} - 1}(\alpha), \alpha\}$ generated by the iterates of $f$ are called cycles. For $\mathbb{Z}[x]$ it is known that cycles of length 1 and 2 occur, and no others. While much is known for extensions to number fields, we concentrate on extending $\mathbb{Z}$ by adjoining reciprocals of primes. Let $\mathbb{Z}[1/p_1, \ldots, 1/p_n]$ denote $\mathbb{Z}$ extended by adding in the reciprocals of the $n$ primes $p_1, \ldots, p_n$ and all their products and powers with each other and the elements of $\mathbb{Z}$. Interestingly, cycles of length 4, called 4-cycles, emerge for polynomials in $\mathbb{Z}\left[1/p_1, \ldots, 1/p_n\right][x]$ under the appropriate conditions. The problem of finding criteria under which 4-cycles emerge is equivalent to determining how often a sum of four terms is zero, where the terms are $\pm 1$ times a product of elements from the list of $n$ primes. We investigate conditions on sets of primes under which 4-cycles emerge. We characterize when 4-cycles emerge if the set has one or two primes, and (assuming a generalization of the ABC conjecture) find conditions on sets of primes guaranteed not to cause 4-cycles to emerge.Comment: 14 pages, 1 figur

Martian Surface Paleotemperatures from Thermochronology of Meteorites

The temporal evolution of past martian surface temperatures is poorly known. We used thermochronology and published noble gas and petrographic data to constrain the temperature histories of the nakhlites and martian meteorite ALH84001. We found that the nakhlites have not been heated to more than 350°C since they formed. Our calculations also suggest that for most of the past 4 billion years, ambient near-surface temperatures on Mars are unlikely to have been much higher than the present cold (<0°C) state

Further evidence for early lunar magnetism from troctolite 76535

The earliest history of the lunar dynamo is largely unknown and has important implications for the thermal state of the Moon and the physics of dynamo generation. The lunar sample with the oldest known paleomagnetic record is the 4.25 billion year old (Ga) troctolite 76535. Previous studies of unoriented subsamples of 76535 found evidence for a dynamo field with a paleointensity of several tens of microteslas. However, the lack of mutual subsample orientation prevented a demonstration that the magnetization was unidirectional, a key property of thermoremanent magnetization. Here we report further alternating field demagnetization on three mutually oriented subsamples of 76535, as well as new pressure remanent magnetization experiments to help rule out shock magnetization. We also describe new 40Ar/39Ar thermochronometry and cosmogenic neon measurements that better constrain the rock's thermal history. Although the rock is unbrecciated, unshocked, and slowly cooled, its demagnetization behavior is not ideal due to spurious remanence acquisition. Despite this limitation, all three subsamples record a high coercivity magnetization oriented in nearly the same direction, implying that they were magnetized by a unidirectional field on the Moon. We find no evidence for shock remanence, and our thermochronometry calculations show no significant reheating events since 4249 ± 12 million years ago (Ma). We infer a field paleointensity of approximately 20–40 μT, supporting the previous conclusion that a lunar dynamo existed at 4.25 Ga. The timing of this field supports an early dynamo powered by thermal or thermochemical core convection and/or a mechanical dynamo but marginally excludes a dynamo delayed by thermal blanketing from radiogenic element-rich magma ocean cumulates

Magnetic tests for magnetosome chains in Martian meteorite ALH84001

Transmission electron microscopy studies have been used to argue that magnetite crystals in carbonate from Martian meteorite ALH84001 have a composition and morphology indistinguishable from that of magnetotactic bacteria. It has even been claimed from scanning electron microscopy imaging that some ALH84001 magnetite crystals are aligned in chains. Alignment of magnetosomes in chains is perhaps the most distinctive of the six crystallographic properties thought to be collectively unique to magnetofossils. Here we use three rock magnetic techniques, low-temperature cycling, the Moskowitz test, and ferromagnetic resonance, to sense the bulk composition and crystallography of millions of ALH84001 magnetite crystals. The magnetic data demonstrate that although the magnetite is unusually pure and fine-grained in a manner similar to terrestrial magnetofossils, most or all of the crystals are not arranged in chains