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I am interested in field extensions of the rationals. About degree 3 extensions there are many refrences including the famous paper of Shanks "The Simplest Cubic Fields". In particular he gave generating polynomial for all such extensions. Later people then classified the parameters of this cubic equation.

I am interested in finding out if any similar results exist in degree 5 and higher cases. Kindly provide refrences if any.

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    $\begingroup$ Giving generating polynomials for a family of number fields that are defined by those very polynomials is not an achievement. What you discover about those number fields might be interesting, though. Please clarify exactly what "similar results" you are asking about. $\endgroup$ – KConrad Feb 7 '14 at 13:13
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    $\begingroup$ This article mentions families of simple fields for cubic, quartic, cubic and sextic fields. $\endgroup$ – Esteban Crespi Feb 7 '14 at 14:44
  • $\begingroup$ Actually, the article also mentions families of quintic fields (a typo, I guess). $\endgroup$ – Dietrich Burde Feb 9 '14 at 13:22
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A. Schwarz, M. Pohst, and F. Diaz y Diaz have computed all quintic number fields of absolute discriminant bounded by $2\cdot 10 ^7$ in case of totally real number fields, and respectively $5\cdot 10^6$ for other signatures. There are $22740$ totally real fields with discriminant less than $2 \cdot 10^7$, by the way. The generating polynomials are given.

Reference: A Table of Quintic number fields.

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