Symbolic Computation Library in pure C
Mathomatic is implemented in C, and may suit your purposes.
Mathomatic™ is a portable, command-line, educational CAS and calculator software, written entirely in the C programming language. It is free and open source software (FOSS), published under the GNU Lesser General Public License (LGPL version 2.1), and has been under continual development since 1986. The software can symbolically solve, simplify, combine, and compare algebraic equations, simultaneously performing generalized standard, complex number, modular, and polynomial arithmetic, as needed. It does some calculus and is very easy to compile/install, learn, and use.
From the developer's manual:
The Mathomatic source code can also be compiled as a symbolic math library that is callable from any C compatible program and is mostly operating system independent.
Unfortunately, the author of this package has passed away, and the software is no longer being maintained. The latest version was archived in GitHub, and the links above have been updated.
Have you taken a look at GAP? From its website:
GAP is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. GAP provides a programming language, a library of thousands of functions implementing algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects. See also the overview and the description of the mathematical capabilities. GAP is used in research and teaching for studying groups and their representations, rings, vector spaces, algebras, combinatorial structures, and more. The system, including source, is distributed freely. You can study and easily modify or extend it for your special use.
According to its Wikipedia page, GAP is implemented in C, and the source code is freely available.
Please look at Axiom -- a general purpose Computer Algebra system. Also you can use Giac -- Giac is a free (GPL) C++ library, it is the computation kernel, it may be used inside other C++ programs.