Can I read Invariant rational functions and a problem of Steenrod on EtoBox?

Invariant rational functions and a problem of Steenrod by Richard G. Swan is a Mathematics article available to read on EtoBox.

What is Invariant rational functions and a problem of Steenrod about?

Let k be a field and let x 1 ..... x, be indeterminates. The symmetric group S, acts on the field K=k(x L, ...,x,) by permuting xl, ...,x,. If G is any subgroup of S,, the field L=K G of elements fixed by G may be described as the field of G-invariant rational functions over k. If G = S n, the field L is a pure transcendental extension of k generated by the elementary symmetric functions. It is an old conjecture of Noether [10] that L is a pure transcendental extension of k for any G. The most important case is that in which k is an algebraic number field. If the conjecture is true in this case, it follows from Hilbert's irreducibility theorem that any group G can be realized as a galois group over k . However, very few results on realizing G as a galois group have been obtained in this way , since the conjecture has proved to be extremely intractible. I will show here that there is good reason for this. The conjecture is false even in the simplest case of a cyclic permutation group.Theorem 1. Let G be the cyclic group of order p acting transitively on the indeterminates x 1 ..... xp. Let L be the fixed fietd Q (xl, ..., xp) ~ where Q is the field of rationals. Then L is not a pure

Who reads Invariant rational functions and a problem of Steenrod?

Author

Richard G. Swan

Published

1969

Language

EN

Field

Mathematics (Physical Sciences)