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I am a undergraduate student with some knowledge of group theory and ring theory. May I ask how should I do to get start to field theory? Could someone tell me what are good to read? Thanks in advance!

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I think the following books on Field and Galois theory are very good.

  1. Galois theory by Ian Stewart. This is the classical one on this topic. It starts with field and galois theory in $\mathbb{C}$, and then develops everything in a general context. If you like to learn from the particular to the general this book is for you.

  2. Galois theory by Joseph Rotman. Another good book on Field and Galois theory. Unlike the first book this starts in the general context as most of the modern books do. Some of its highlights are the classical Hilbert's 90 theorem and the normal basis theorem.

  3. Galois theory by Steven Weintraub. This book is similar to Rotman's books, but it has some extra theorems and topics like infinite galoisian extensions (which makes use of some stuff of general topology).

  4. Field theory by Steven Roman. This is for sure one of the most complete books on Field and Galois theory. The author like Rotman and Weintraub approaches the course in an abstract setting but Roman goes further since for example in the topic of Galois theory the author develops the theory of "Galois connections". Also in this book like in Weintraub's book it is studied infinite galoisian extensions.

Needless to say, every one of these books covers the basic stuff of Field and Galois theory.

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Kaplansky's Fields and Rings is concise and very good, and I'm a fan of the exercises that he gives.

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