費馬最後定理在其聞名後一兩百年，仍是一個很大的謎題。雖然現今已被完全證明，不過這篇文章則以探討費馬最後定理在十九世紀的發展結果及背後的一些理論，也就是Kummer所做的研究為主。內容涵括伯努力數、分圓體、理想子環、正則質數，以及Kummer引理，並參考這些已知的定理和一些代數數論的理論，改寫Kummer對於費馬最後定理的證明。主要的證明方式為將定理區分為兩個狀況：其次方不整除變數，以及次方整除唯一一個變數。前者使用單純的反證法證明，後者則使用無窮遞減法證明矛盾。

The Fermat’s Last Theorem was first conjectured in 17th Century, since than, many mathematicians tried hard to solve it. Although it has been totally proved by now, but this article will focus on the development of Fermat’s Last Theorem in 19th Century and its background theories, that is, the research which made by Kummer. Includes Bernoulli Numbers, Cyclotomic Field, Ideals, Regular Primes, and Kummes’s Lemma, and use these theorems we’ve learned and some algebraic number theory to help us rewrite the Kummer’s prove for Fermat’s Last Theorem. The main idea of the prove is to separate the theorem into two case: The power doesn’t divides any of the variables; and the power divides one of the variables. Both of them are proved by contradiction, but the later one is also using Infinite Descent.

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