MINISTRY OF EDUCATION
AND TRAINING
VIETNAM ACADEMY
OF SCIENCE AND TECHNOLOGY
GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY
Le Khac Nhuan
GALOIS COHOMOLOGY AND ITS
APPLICATIONS IN THE EMBEDDING PROBLEM
MASTER THESIS IN MATHEMATICS
Hanoi - 2021
BỘ GIÁO DỤC
VÀ ĐÀO TO
VIỆN HÀN LÂM
KHOA HỌC VÀ CÔNG NGHỆ VN
HỌC VIỆN KHOA HỌC VÀ CÔNG NGHỆ
Khắc Nhuận
ĐỐI ĐỒNG ĐIỀU GALOIS
VÀ ỨNG DỤNG VÀO BÀI TOÁN NHÚNG
Chuyên ngành: Đại số và Lý thuyết số
số: 8460104
LUẬN VĂN THẠC NGÀNH TOÁN HỌC
NGƯỜI HƯỚNG DẪN KHOA HỌC:
PGS. TS. Nguyễn Duy Tân
Nội - 2021
MINISTRY OF EDUCATION
AND TRAINING
VIETNAM ACADEMY
OF SCIENCE AND TECHNOLOGY
GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY
Le Khac Nhuan
GALOIS COHOMOLOGY AND ITS
APPLICATIONS IN THE EMBEDDING PROBLEM
Major: Algebra and Number theory
Code: 8460104
MASTER THESIS IN MATHEMATICS
SUPERVISOR:
Assoc. Prof. Nguyen Duy Tan
Hanoi - 2021
1
Declaration
I declare that this thesis titled ”Galois cohomology and its applications in the em-
bedding problem” and the work presented in it are my own. Wherever the work of
others are involved, every effort is made to indicate this clearly, with due reference
to the literature. I confirm that this thesis has not been previously included in a the-
sis or dissertation submitted for a degree or any other qualification at this graduate
university or any other institution. I will take responsibility for the above declaration.
Hanoi, 20th September 2021
Signature of Student
Le Khac Nhuan
2
Acknowledgement
First of all, I would like to express my deepest gratitude to Assoc. Prof. Nguyen
Duy Tan was willing to supervise me despite my impoverished undergraduate mathe-
matical background. Thanks to his topic suggestion and his supervision, I have learnt
so many things about Galois cohomology in particular and mathematics in general.
His guidance helped me all the time of writing this thesis. I could not have imagined
having a better supervisor for my MSc study.
Secondly, I would like to thank my lecturers, especially Dr. Nguyen Dang Hop
and Dr. Nguyen Chu Gia Vuong, for many valuable mathematical lessons and in-
sights. I would like also to thank the Graduate University of Science and Technology
and the Institute of Mathematics for providing a fertile mathematical environment.
Special thanks goes to Nguyen Van Quyet and Quan Thi Hoai Thu who all helped me
in numerous ways during my study in Hanoi. To Quyet in particular I owe you one for
being a supportive friend as well as keeping me feel accompanied and conversations
on various issues besides mathematics during my leisure times.
I would like also to thank my girlfriend Pham Thi Minh Tam, who has been by
my side throughout my MSc, for all her love and support.
Last but not least, my thanks goes to my parents and siblings for always believing
in me and encouraging me to follow my dreams.