Degree 12 2-adic fields with automorphism group of order 4

Awtrey, Chad; Barkley, Brett; Miles, Nicole; Strosnider, Erin

doi:10.1216/rmj-2015-45-6-1755

Cited by 2 publications

(5 citation statements)

References 16 publications

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“…• T 1 is stabilized by the subgroup H 1 generated by {(1,2), (3,4), (3,4,5,6,7,8,9,10,11,12,13,14,15)}.…”

Section: Three Resolvent Polynomialsmentioning

confidence: 99%

“…• T 3 is stabilized by the subgroup H 3 generated by {(1,2), (3,4,5), (4,6,7,8,9,10,11,12,13,14,15), (3,6,7,8,9,10,11,12,13,14,15)}. note, the subscript gives the degree of the resolvent polynomial.…”

Section: Three Resolvent Polynomialsmentioning

confidence: 99%

“…Otherwise, the situation is more complicated, with no practical general algorithm currently available. However, several researchers have developed ad hoc techniques that depend on both the degree of f and the prime p ( [2,3,4,5,6,7,11,12,13]).…”

Section: Introductionmentioning

confidence: 99%

“…6 + 47x 5 + 60x 4 + 80x 3 + 110x + 67x 15 + 90x 14 + 15x 13 + 70x 12 + 110x 11 + 60x 10 + 85x 9 + 110x 8 +…”

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On Galois Groups of Degree 15 Polynomials

Awtrey¹,

Mazur²,

Rodgers³

et al. 2015

Int. J. of Pure and Appl. Math.

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We discuss the construction and factorization pattern of several resolvent polynomials that are useful for computing Galois groups of degree 15 polynomials. As an application, we develop an algorithm for computing the Galois group of a degree 15 polynomial defined over the 5-adic numbers. This algorithm is of interest since it uses substantially fewer resolvents than the traditional method for computing Galois groups.

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“…• T 1 is stabilized by the subgroup H 1 generated by {(1,2), (3,4), (3,4,5,6,7,8,9,10,11,12,13,14,15)}.…”

Section: Three Resolvent Polynomialsmentioning

confidence: 99%

Section: Three Resolvent Polynomialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…6 + 47x 5 + 60x 4 + 80x 3 + 110x + 67x 15 + 90x 14 + 15x 13 + 70x 12 + 110x 11 + 60x 10 + 85x 9 + 110x 8 +…”

unclassified

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On Galois Groups of Degree 15 Polynomials

Awtrey¹,

Mazur²,

Rodgers³

et al. 2015

Int. J. of Pure and Appl. Math.

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“…Likewise, when p = n, the situation has been solved since the early 1970s [Amano 1971;Jones and Roberts 2006]. The difficult cases where p | n and n is composite have been dealt with on a case-by-case basis for low degrees n and small primes p. Jones and Roberts [2004;2006; have classified the cases where n ≤ 10, and the case of degree 12 is dealt with in [Awtrey 2012;Awtrey and Shill 2013;Awtrey et al ≥ 2015a;≥ 2015b].…”

Section: Introductionmentioning

confidence: 99%

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The calculation of the probability of correct selection (PCS) shows how likely it is that the populations chosen as "best" truly are the top populations, according to a well-defined standard. PCS is useful for the researcher with limited resources or the statistician attempting to test the quality of two different statistics. This paper explores the theory behind two selection goals for PCS, G-best and d-best, and how they improve previous definitions of PCS for massive datasets. This paper also calculates PCS for two applications that have already been analyzed by multiple testing procedures in the literature. The two applications are in neuroimaging and econometrics. It is shown through these applications that PCS not only supports the multiple testing conclusions but also provides further information about the statistics used.ENHANCING MULTIPLE TESTING 185 selection of the top populations. After an experiment, one can use these methods to calculate the probability that the researcher has found the actual top t populations.If our previous toy example was an actual experiment, we might have a need to find the top, say, one statistic, but we have the resources to study two. We would then use G-best selection, where we choose a fixed amount of populations, t + G, that contains the top t statistics. On the other hand, if we simply needed the populations to be within a certain threshold of quality, we would use d-best selection. In d-best selection, we are finding a random number of populations, say r , which contains populations that are within a certain distance d from the top t populations. The number r is determined by an interval of prespecified length d.Definition 2.1. Let s be the set of the indices corresponding to the top t +G statistics for some prespecified G. Let A t be a set of indices of the top t parameters. ThenA set s that satisfies CS G,t is called G-best, and the probability that we have chosen a G-best set is denoted by P(CS G,t ). 190ERIN IRWIN AND JASON WILSON t 1 2 3 4 P(CS 0,t ) = P( 0 CS t ) 0.11 0.02 0.00 0.00 P(CS 2,t ) 0.33 0.11 0.03 0.01 P(CS 4,t ) 0.49 0.22 0.08 0.03 P(CS 6,t ) 0.60 0.33 0.15 0.06 P( 0.5 CS t ) 0.19 0.04 0.05 0.03 P( 1 CS t ) 0.24 0.19 0.35 0.29P(CS 0,10 ) = 0.13 P( 0 CS 10 ) = 0.13 P(CS 0,10 ) = 0.71 P( 0 CS 10 ) = 0.71 P(CS 1,9 ) = 0.29 P( 0.5 CS 10 ) = 0.32 P(CS 1,9 ) = 0.98 P( 0.5 CS 10 ) = 0.71 P(CS 2,8 ) = 0.53 P( 1 CS 10 ) = 0.52 P(CS 2,8 ) = 1.00 P( 1 CS 10 ) = 0.95 P(CS 3,7 ) = 0.74 P( 1.5 CS 10 ) = 0.78 P(CS 3,7 ) = 1.00 P( 1.5 CS 10 ) = 0.95 P(CS 4,6 ) = 0.92 P( 2 CS 10 ) = 0.86 P(CS 4,6 ) = 1.00 P( 2 CS 10 ) = 0.97 P(CS 5,5 ) = 1.00 P( 2.5 CS 10 ) = 0.93 P(CS 5,5 ) = 1.

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Degree 12 2-adic fields with automorphism group of order 4

Cited by 2 publications

References 16 publications

On Galois Groups of Degree 15 Polynomials

On Galois Groups of Degree 15 Polynomials

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