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# Solvable Points on Projective Algebraic Curves

We examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus $g$, when $g$ is at least $40$, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus $0$, $2$, $3$ or $4$ defined over any field has a solvable point. Finally we prove that every genus $1$ curve defined over a local field of characteristic zero with residue field of characteristic $p$ has a divisor of degree prime to $6p$ defined over a solvable extension.
 MSC Classifications: 14H25 - Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx] 11D88 - $p$-adic and power series fields