Automorphism Polynomials in Cyclic Cubic Extensions

✍ Scribed by Robin J. Chapman

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 254 KB
• Volume: 61
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0149

✦ Synopsis

We provide simple proofs of the main results in the paper by Patrick Morton, ``Characterizing Cyclic Cubic Extensions by Automorphism Polynomials'' (J. Number Theory 49 (1994), 183 208), avoiding the use of computer algebra.

1996 Academic Press, Inc.

1. Introduction

In [2] Morton considers cyclic cubic extensions FÂ} of fields. If _ is a generator of the Galois group of this extension, he shows that if % # F and _(%)=% 2 +u (u # }) then, provided char }{2, u=&(1Â4)(s 2 +7) where s # } and the field F is determined by s. He also proves a theorem showing when two values of the parameter s # } give the same extension field and develops an analogue of Kummer theory, not assuming that } contains the third roots of unity, describing the exponent 3 Abelian extensions of }. However, some of his proofs rely on extensive Mathematica computation, and he asks if there are less computationally demanding proofs.

In this paper I relate Morton's analogue of Kummer theory to classical Kummer theory and Artin Schreier theory. In this way I provide much simpler proofs of Morton's results.

2. THE GENERIC CASE

We first deal with the case where char }{3 and } does not have a primitive cube root of unity. In this case let }$=}() where is a primitive cube root of unity. The extension }$Â} is quadratic; let { be its non-trivial automorphism. If * # }$ for convenience we shall write * ={(*), N(*)=** , and T(*)=*+* .

If FÂ} is a cyclic cubic extension, then F $=F(`) is a cyclic cubic extension of }$ and a cyclic sextic extension of }. Conversely each cyclic sextic article no. 0149