Iwasawa theoretical studies of Galois groups over algebraic number fields with restricted ramifications

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The Galois groups, which express the symmetry of the solutions of algebraic equations (roots of polynomials with integral coefficients), become the objects expressing a number of interesting properties (for example, uniqueness of prime factorization) of algebraic integers by considering the restricted ramifications. For a certain unsolved conjecture in Iwasawa theory, which concerns the prime factorization in higher dimensions, this research clarifies the non-commutative structures of certain Galois groups with restricted ramifications, and gave new affirmative examples of the conjecture by considering the non-commutative properties which are immanent in prime numbers.