The regular heptagon is the first regular polygon that compass and straightedge cannot construct. Not the first that is difficult, the first that is impossible, in the strict sense that no finite sequence of the permitted operations can produce it.
The governing result is the Gauss-Wantzel theorem: a regular polygon with n sides is constructible with compass and straightedge if and only if n is a power of 2 multiplied by a product of distinct Fermat primes. A Fermat prime has the form 2 to the power 2 to the power k, plus one, and only five are known: 3, 5, 17, 257 and 65537. Seven is prime, but it is not a Fermat prime and it is not a power of two. The heptagon fails the condition.
Gauss supplied the sufficient half of the theorem and published it in the Disquisitiones Arithmeticae of 1801, having recorded on 30 March 1796, a month before his nineteenth birthday, that the regular 17-gon is constructible. That was the first advance on the classical list of constructible polygons in over two thousand years, and he is said to have wanted a 17-gon on his gravestone; the monument at Brunswick carries a seventeen-pointed star instead, the mason having observed that a 17-gon would be mistaken for a circle. Gauss asserted the necessary half without proving it. Pierre Wantzel supplied that proof in 1837, and it is Wantzel's contribution that converts the statement from a criterion for success into a demonstration of impossibility.
The algebra behind it is concrete. Every point reachable by compass and straightedge comes from intersecting lines and circles, and each such step solves at worst a quadratic, so every constructible length has degree a power of 2 over the rationals. Constructing a regular heptagon requires the quantity 2 cos of 2 pi over 7, whose minimal polynomial is the cubic x cubed plus x squared minus 2x minus 1, irreducible over the rationals. Degree 3 is not a power of 2. There is nothing further to be said: the number lies out of reach of the tools.