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In 2020, Alabdali and Byott described the Hopf-Galois structures arising on Galois field extensions of squarefree degree. Extending to squarefree separable, but not necessarily normal, extensions L / K is a natural next step. One must consider now the interplay between two Galois groups G = Gal (E / K) and G ′ = Gal (E / L) , where E is the Galois closure of L / K. In this paper, we give a characterisation and enumeration of the Hopf-Galois structures arising on separable extensions of degree pq where p and q are distinct odd primes. This work includes the results of Byott and Martin-Lyons who do likewise for the special case that p = 2 q + 1. [ABSTRACT FROM AUTHOR]

An n × n matrix pattern is said to be spectrally arbitrary over a field F provided for every monic polynomial p ( t ) of degree n , with coefficients from F , there exists a matrix with entries from F , in the given pattern, that has characteristic polynomial p ( t ) . Let E ⊆ F ⊆ K be an extension of fields. It is natural to ask whether a pattern that is spectrally arbitrary over F must also be spectrally arbitrary over E or K . In this article it is shown that if F is dense in K and K is a complete metric space, then any spectrally arbitrary or relaxed spectrally arbitrary pattern over F is relaxed spectrally arbitrary over K . It is also established that if E is an algebraically closed subfield of a field F , then any spectrally arbitrary pattern over F is spectrally arbitrary over E . The 2 n Conjecture and the Superpattern Conjecture are explored over fields other than the real numbers. In particular, examples are provided to show that the Superpattern Conjecture is false over the field with 3 elements. [ABSTRACT FROM AUTHOR]

In 2021, Marco Besier and the first author introduced the concept of rationalizability of square roots to simplify arguments of Feynman integrals. In this work, we generalize the definition of rationalizability to field extensions. We then show that the rationalizability of a set of quadratic field extensions is equivalent to the rationalizability of the compositum of the field extensions, providing a new strategy to prove rationalizability of sets of square roots of polynomials. [ABSTRACT FROM AUTHOR]

We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions L / K of squarefree degree n. If E / K is the normal closure of L / K then G = Gal (E / K) can be viewed as a permutation group of degree n. We show that G has derived length at most 4, but that many permutation groups of squarefree degree and of derived length 2 cannot occur. We then investigate in detail the case where n = p q where q ≥ 3 and p = 2 q + 1 are both prime. (Thus q is a Sophie Germain prime and p is a safeprime). We list the permutation groups G which can arise, and we enumerate the Hopf-Galois structures for each G. There are six such G for which the corresponding field extensions L / K admit Hopf-Galois structures of both possible types. [ABSTRACT FROM AUTHOR]

Abstract: In a previous piece of work it has been shown that the exponential stability of a linear time-varying (LTV) system can be evaluated using new definitions of the poles of such a system. The latter are given by a fundamental set of roots of the skew polynomial which defines the autonomous part of the system. Such a set may not exist over the initial field of definition of the coefficients of the system, but can exist over a suitable field extension . It is shown here that conditions for stability can also be obtained using linear factors of the polynomial over another field extension which may be smaller: . The roots of these factors are called the quasi-poles of the system. The necessary condition for system stability, expressed in function of these quasi-poles, is more restrictive than the one involving a fundamental set of roots. [Copyright &y& Elsevier]