## CARLITZ EXTENSIONS

### BibTeX

@MISC{Conrad_carlitzextensions,

author = {Keith Conrad},

title = {CARLITZ EXTENSIONS},

year = {}

}

### OpenURL

### Abstract

The ring Z has many analogies with the ring Fp[T], where Fp is a field of prime size p. For example, for nonzero m ∈ Z and M ∈ Fp[T], the residue rings Z/(m) and Fp[T]/M are both finite. The unit groups Z × = {±1} and Fp[T] × = F × p are both finite. Every nonzero integer can be made positive after multiplication by a suitable sign ±1, and every

### Citations

226 |
Number theory
- Borevich, Shafarevich
- 1966
(Show Context)
Citation Context ...feld in the 1970s). 2. Carlitz polynomials For each M ∈ Fp[T ] we will define the Carlitz polynomial [M](X) with coefficients in Fp[T ]. Our definition will proceed by recursion and linearity. Define =-=[1]-=-(X) := X and For n ≥ 2, define [T ](X) := X p + T X. [T n ](X) := [T ]([T n−1 ](X)). Example 2.1. [T 2 ](X) = [T ]([T ](X)) = (Xp +T X) p +T (Xp +T X) = Xp2 +(T p +T )Xp + T 2X. For a general polynomi... |

162 | Basic structures of function field arithmetic - Goss - 1996 |

65 | Number Theory in Function Fields - Rosen - 2002 |

51 | On certain functions connected with polynomials in a Galois field - Carlitz - 1935 |

37 |
Explicit class field theory for rational function fields
- Hayes
- 1974
(Show Context)
Citation Context ...eads ΦM(0) = 1. The Kronecker-Weber theorem says every finite abelian extension of Q lies in a cyclotomic extension Q(µm). There is an analogue of the Kronecker-Weber theorem for Fp(T ), due to Hayes =-=[4]-=-. It says every finite abelian extension of Fp(T ) lies in some F p d(T, ΛM, Λ 1/T n) for some d ≥ 1, n ≥ 1, and M ∈ Fp[T ], where Λ 1/T n is the set of roots of the Carlitz polynomial [1/T n ](X) bui... |

20 |
A set of polynomials
- Carlitz
- 1940
(Show Context)
Citation Context ...[M](X) to Fp(T ) will yield a field extension of Fp(T ) whose Galois group is isomorphic to (Fp[T ]/M) × instead of (Z/(m)) × . The polynomials [M](X) and their roots were first introduced by Carlitz =-=[2, 3]-=- in the 1930s. Since Carlitz gave his papers unassuming names (look at the title of [3]), their relevance was not widely recognized until being rediscovered several decades later (e.g., in work of Lub... |

17 |
Certain quantities transcendental over GF(p n ,x
- Wade
- 1941
(Show Context)
Citation Context ...ies in Definition 6.1, just like π doesn’t appear in the definition of the usual exponential series. The value ξ is a characteristic p analogue of 2πi. It is not in Fp((1/T )), just as 2πi ̸∈ R. Wade =-=[8]-=- proved ξ is transcendental over Fp(T ), which is analogous to 2πi being transcendental over Q. As a function on Fp((1/T )), the formal power series for eC(X) is an “entire function”: it converges eve... |

6 | Algebra, 3rd revised Ed - LANG - 2002 |

4 |
On Artin’s conjecture for Carlitz modules
- Hsu
- 1997
(Show Context)
Citation Context ... has two polynomial analogues: when does a polynomial in Fp[T ] generate infinitely many of the groups (Fp[T ]/π)× or infinitely many of the modules C(Fp[T ]/π)? Both questions have good answers; see =-=[5]-=- and [7, Chap. 10]. In another direction, for any prime p ̸= 2 the groups (Z/(pk))× are cyclic for all k ≥ 1, but the groups (Fp[T ]/πk)× are not cyclic for any k ≥ 4. (It is cyclic for k = 2 if deg π... |