本書是根據美國著名數學傢S.Lang教授多年來在多所知名大學為大學生舉辦的教學專題講座的講稿編寫而成的。主要包括：素數分布猜想，abc猜想，局部可積嚮量場的整體積分，分析中的逼近定理，Bruhat-Tits度量空間及其不動點定理，調和多項式和對稱多項式的嚮量空間的結構等數學問題及應用。

本書敘述清楚，語言生動，覆蓋麵廣，寓意深刻。它可使具有微積分和綫代數基礎知識的讀者迅速地接觸到多方麵的現代數學題材，激發他們對數學的濃厚興趣，提高數學素質。本書使用的精湛教學藝術和卓越外理技巧於廣大數學教師和數學工作者也有很大的啓發性。

本書可作為理工科大學生和研究生的數學課外參考書討論班輔助材料，也可供數學教師和數學愛好者參考。

Serge Lang (May 19, 1927 – September 12, 2005) was a French-born American mathematician. He is known for his work in number theory and for his mathematics textbooks, including the influential Algebra. He was a member of the Bourbaki group.

Lang was born in Paris in 1927, and moved with his family to California as a teenager, where he graduated in 1943 from Beverly Hills High School. He subsequently graduated from the California Institute of Technology in 1946, and received a doctorate from Princeton University in 1951. He held faculty positions at the University of Chicago and Columbia University (from 1955, leaving in 1971 in a dispute). At the time of his death he was professor emeritus of mathematics at Yale University.