Links
Books, LectureS
Survey, Shpilka & Yehudayoff (FnTTCS, 2010).
Lower bound surveys in link.
PIT Surveys 1, 2, Saxena.
Book, Modern Computer Algebra, von zur Gathen & Gerhard.
Book, Finite Fields, Lidl & Niederreiter.
Book, Algorithmic Number Theory (Lattices, Number Fields, Curves and Cryptography), Edited by Joseph P. Buhler and Peter Stevenhagen.
Book, Algorithmic Number Theory, Bach & Shallit.
Course, Topics in Complexity, Kayal & Saha.
Course, Algebra and Computation, Madhu Sudan.
Course, Computational Number Theory and Algebra, Nitin Saxena.
Online Stuff
Arithmetic Circuit Complexity, Computational number theory, algebra.
Open problems, Adleman & McCurley.
History: Algebra.
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``Algebra is generous; she often gives more than is asked of her.''
-- Jean d’Alembert (1717–1783).