# Research and Publications

### Background and Current Interests

I work on problems arising from the interaction between algebraic number theory, Galois theory and group theory. In order to explain where the problems I'm interested in come from, a little history is in order. In the nineteenth century, mathematicians realized that certain problems involving integers could be solved or understood at a deeper level by working in more general number systems and the study of these systems (now called rings of integers) became the main goal of classical algebraic number theory. One of the early discoveries about these rings was that they no longer necessarily possess the unique factorization property which is a useful property of the ordinary integers. Understanding when these rings possess this property became an important area of investigation.

It turns out that one can associate a finite abelian group (called the class group) to each such ring which characterizes the failure (or not) of unique factorization. In the early part of the twentieth century, it was discovered that the class group can be thought about in a completely different way: as the Galois group of a certain special field extension of the field of fractions of the ring of integers, the so called Hilbert class field extension. (In fact, this is just one more small part of a branch of number theory called class field theory which, in one formulation, characterizes the abelian extensions of number fields in terms of more general objects called ray class groups.)

It was observed that iterating the construction of this special extension to obtain a tower of fields was directly related to the problem of determining if one can always embed a given ring of integers in a larger ring of integers guaranteed to possess the unique factorization property. For a time it was believed that such an embedding would always exist, but in 1964 this was shown to be false in a landmark paper by Golod and Safarevic. This paper and subsequent refinements provided a simple sufficient criterion for when a given pro-p group is infinite. Applied to the Galois groups associated to certain towers, it could be shown that the towers were infinite and hence that the embedding problem for the base field was not solvable.

My work largely centers on questions about the structure of the Galois groups associated to such towers and other maximal field extensions with restricted ramification. What kinds of groups can arise? Can they be classified in some way? Given a particular group, can one say anything about how frequently this group should occur? I'm also interested more generally in certain special classes of groups (finite and infinite) including p-groups, groups of deficiency zero, branch groups and mild pro-p groups. Some other topics in which I have varying degrees of interest include: root discriminant bounds for number fields, computational algebra (in particular computational group theory and number theory), the inverse problem in Galois theory and error-correcting codes.

In recent years I've supervised several research projects, involving small groups of between 2 and 4 undergraduate students, on various topics including properties of the trellis complexity of codes, De Bruijn sequences and sequences defined via recurrences over various algebraic structures (such as Z_n), and questions about permutation groups that arise in the English art of change ringing. (I've actually been change ringing here in the USA, but not for a while due to the absence of nearby towers.)

In general, I'm willing to supervise projects in algebra, number theory and other parts of discrete mathematics. If you're a W&L student and are interested in working on something then there are various possibilities. Stop by and talk to me some time!

### Publications and Preprints

#### In Preparation:

• Schur σ-groups of small prime power order.

• Heuristics for p-class towers of real quadratic fields, joint with N. Boston and F. Hajir.

#### Published:

• Galois groups of iterates of some unicritical polynomials, (pdf)
M. R. Bush, W. Hindes, N. Looper,
accepted by Acta Arithmetica.
• Heuristics for p-class towers of imaginary quadratic fields, (pdf)
N. Boston, M. R. Bush, F. Hajir,
Math. Annalen, 368 (2017), 633--669.
• 3-class field towers of exact length 3, (pdf)
M. R. Bush, D. C. Mayer,
J. Number Theory, 147 (2015), 766--777.

• Different Partners, Different Places: Mathematics applied to the construction of four-couple folk dances, (pdf)
M. R. Bush, G. M. Roodman,
Journal of Mathematics and the Arts, 7 (2013), 17--28.
• Mild 2-relator pro-p-groups, (pdf)
M. R. Bush, J. Gärtner, J. Labute, D. Vogel,
New York J. Math., 17 (2011), 281--294.
Published version is freely available online from the journal.

• An irreducibility lemma, (pdf)
M. R. Bush, F. Hajir,
J. Ramanujan Math. Soc., 23, No. 1 (2008), 1--9.

• Mild pro-p Galois groups with 4 generators, (pdf)
M. R. Bush, J. Labute,
J. Algebra, 308 (2007), 828--839.

• Maximal unramified 3-extensions of imaginary quadratic fields and SL_2(Z_3), (pdf)
L. Bartholdi, M. R. Bush,
J. Number Theory, 124 (2007), 159--166.

• Maximal unramified 3-extensions of imaginary quadratic fields and SL_2(Z_3), extended abstract of talk (pdf)
M. R. Bush,
Oberwolfach Reports for 2006.

• Computation of the Galois groups associated to the 2-class towers of some quadratic fields, (pdf)
M. R. Bush,
J. Number Theory, 100 (2003), 313--325.

• Computing left Kan extensions,
M. R. Bush, M. Leeming, R. F. C. Walters,
J. Symbolic Comput. 35 (2003), 107--126.
Available online from JSC.

• Integral equation approximation for inhomogeneous fluids: functional optimization,
M. R. Bush, M. Booth, A. D. J. Haymet, A. G. Schlijper,
Molecular Physics, Vol. 95, No. 3 (1998), 601--619.
• ### Theses

• p-class towers of imaginary quadratic fields,
Ph.D. Dissertation, University of Illinois at Urbana-Champaign (May 2004).

• The Todd-Coxeter procedure and its generalisations,
Honours Thesis, University of Sydney (November 1996).