The Chebyshev Exponent

The analogy between the nth power function and the nth Chebyshev polynomial is pursued, leading to consideration of Chebyshev radicals as analogous to ordinary radicals and Chebyshev exponents to ordinary exponents, and the cosine and hyperbolic cosine as analogs of the exponential function. We then discuss solving polynomial equations in Chebyshev radicals, and apply this to the construction of unramified extensions of quadratic number fields.