### Abstract

Let A = double-struck F_{q}[T], where double-struck F_{q} is a finite field, let Q = double-struck F_{q}(T), and let F be a finite extension of Q. Consider φ a Drinfeld A-module over F of rank r. We write r = hed, where E is the center of D :=End _{F{combining macron}}(φ)⊗Q, e = [E : Q] and d = [D : E]1/2. For m ∈ A, let F (φ[m]) be the field obtained by adjoining to F the m-division points φ[m] of φ, and let F (φ[m])′ be the subfield of F (φ[m]) fixed by the scalar elements of Gal(F (φ[m])/F) ⊆ GL_{r}(A/mA). In this paper, when r ≥ 3 and h ≥ 2, we study the splitting of the primes ℘ of F of degree x in the fields F (φ[m])′ and obtain an asymptotic formula which counts them.

Original language | English |
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Pages (from-to) | 211-221 |

Number of pages | 11 |

Journal | Houston Journal of Mathematics |

Volume | 42 |

Issue number | 1 |

Publication status | Published - 2016 Jan 1 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Virdol, C. (2016). Drinfeld modules and subfields of division fields.

*Houston Journal of Mathematics*,*42*(1), 211-221.