This contributed volume presents recent advances as well as new directions in number theory and its applications. Algebraic and analytic number theory are the main focus with chapters showing how these areas are rapidly evolving. By gathering authors from over seven countries, readers will gain an international perspective on the current state of research as well as potential avenues to explore. Specific topics covered include:

• Algebraic Number Theory
• Elliptic curves and Cryptography
• Hopf Galois theory
• Analytic and elementary number theory and applications New Frontiers in Number Theory and Applications will appeal to researchers interested in gaining a global view of current research in number theory.

Presents recent advances as well as new directions in number theory and its applications Shows the rapid evolution of the field with a particular focus on analytic and algebraic number theory Offers an international perspective on the current state and future directions of these research areas

Autorentext
Jordi Guàrdia is an associate professor in the Math Department at Universitat Politècnica de Catalunya-Barcelona Tech. His research has been focused in arithmetic geometry and computational number theory. Jointly with J. Montes and E. Nart, he has been involved in the development of Montes algorithm and its theoretical and computational applications, including the Magma package "+Ideals". They are now working in higher rank valuation theory and the interaction between number theory and singularities theory. He is one of the founders of the biannual conference" Jornadas de Teoria de Numeros", and is an active member of the "Seminaride Teoria de Nombres de Barcelona".Nicüor Minculete is an associate professor and vice dean at the Faculty of Mathematics and Computer Science, Transilvania University of Bräov, Romania. Nicüor Minculete's degrees earned: Mathematics at University of Bucharest (1994): Ph.D. in Mathematics at Institute of Mathematics of the Romanian Academy (2012). Research interests: Mathematical Inequalities and its Applications; Number Theory; Euclidean geometry. Member of the Editorial Board at the following journals: European Journal of Mathematics and Applications, International Journal of Geometry, Bulletin of the Transilvania University of Brasov, General Mathematics, Octogon Mathematical Magazine. He has published 100 research papers.

Diana Savin is an associate professor at the Faculty of Mathematics and Computer Science, Transilvania University of Bräov, Romania. Diana Savin graduated from the Faculty of Mathematics and Computer Science of the University of Bucharest in 1996. She obtained the PhD degree in Mathematics at Ovidius University of Constanta, Romania, in 2004, with a thesis about Diophantine equations. Diana Savin works in number theory. Her research directions are algebraic number theory (especially ramification theory in algebraic number fields), associative algebras, computational numbertheory and combinatorics. She wrote several research articles on these areas. Alone or jointly with V. Acciaro, M. Taous and A. Zekhnini, she has been studying quaternion algebras over some algebraic number fields, using ramification theory in algebraic number fields.

Montserrat Vela is an associate professor of the Math Department at Universitat Politècnica de Catalunya-Barcelona Tech. She has worked mainly in the inverse problem in Galois theory, Nowadays, her research is focused on Hopf-Galois theory, having published some relevant papers jointly with T. Crespo and A. Rio. She is an active member of the "Seminari de Teoria de Nombres de Barcelona".

Abdelkader Zekhnini is an associate professor at the Mohammed Premier University, Sciences Faculty, Department of Mathematics, Oujda, Morocco. He obtained the PhD degree in Mathematics in 2014, at the same university with a thesis on capitulation theory and Hilbert class field towers. Before, he was a mathematics teacher in high schools. Abdelkader Zekhnini works in many directions of number theory as algebraic number theory (especialy capitulation theory), Hilbert class field towers, Iwasawa theory, commutative algebra (Integer valued polynomials, Polya fields) and associative algebras. He wrote several research papers on these areas, over 40, published in Web of Science indexed journals.

Inhalt
Chapter. 1. Survey Number Theoretic Transform Algorithm over a polynomial ring and its application.- Chapter. 2. On the Hilbert 2-class field towers of the layers of some cyclotomic Z2- extensions.- Chapter. 3. Some remarks on the coefficients of cyclotomic polynomials.- Chapter. 4. Partitions enumerated by self-similar sequences.- Chapter. 5. On the exact divisibility by 5 of the class number of some pure metacyclic fields.- Chapter. 6. On monogenity of certain number fields defined by a trinomial xpr + ax + b.- Chapter. 7. Irreducible Factors of a Polynomial and Extensions of Valuations.- Chapter. 8. Lengths and Class Numbers.- Chapter. 9. On Extended korder Fibonacci and Lucas Numbers via DGC Numbers.- Chapter. 10. Generalizations of Stirling-like and Bell-like numbers.- Chapter. 11. The complex multiplication method for genus 3 curves.- Chapter. 12. An optimal Chevalley-Warning theorem over the field of two elements.- Chapter. 13. New perspectives of the power-commutator structure: coclass trees of CF-groups and related BCF-groups.- Chapter. 14. The Euler-Riemann zeta function with even arguments in terms of binomial coefficients.- Chapter. 15. Some properties of a type of entropy of an ideal and the divergence of two ideals.- Chapter. 16. A simple and self-contained proof for the Lindemann-Weierstrass theorem.- Chapter. 17. On the 2-class number of some real cyclic quartic number fields II.- Chapter. 18. On the p -isogenies of elliptic curves with multiplicative reduction over quadratic fields.- Chapter. 19. The Hopf-Galois structures of the 3 and 6-division points of the lemniscate curve.