The Brauer-manin Obstruction to the Hasse Principle

Acknowledgements My greatest gratitude goes out to my supervisor Dr Daniel Chan. He helped me pick the topic for this thesis which turned out to be extremely interesting, helped me with the writing of this thesis and most importantly never refused to spend time with me to help me learn mathematics. I thank him deeply for this. I would to also like thank my high school maths teacher Ian Denton for getting me interested in mathematics to begin with. Similarly, a special thanks goes to Dr Ian Doust who, in response to my email requesting permission to avoid doing any algebra courses replied: " I'm not going to let you out of here with a pure maths degree without seeing some proper group theory! ". I hope this thesis shows that I have learnt the basics of group theory, and Ian will let me graduate. Thank you to Kenneth and Hugo, for helping me throughout this year, and thank you to Anna for being supportive and putting up with me. Special thanks also goes to my family.