#### Glasnik Matematicki, Vol. 46, No.1 (2011), 11-14.

*k*TH POWER RESIDUE CHAINS OF GLOBAL FIELDS

### Su Hu and Yan Li

Department of Mathematical Sciences, Tsinghua University,
Beijing 100084, China

*e-mail:* `hus04@mails.tsinghua.edu.cn`

*e-mail:* `liyan_00@mails.tsinghua.edu.cn`

**Abstract.** In 1974, Vegh proved that if *k* is a prime and *m* a positive integer,
there is an *m* term permutation chain of *k*th power residue for
infinitely many primes (E.Vegh, *k*th power residue chains, J. Number
Theory 9 (1977), 179-181). In fact, his proof showed that
*1,2,2*^{2}, ..., 2^{m-1} is an *m* term permutation chain of *k*th
power residue for infinitely many primes. In this paper, we prove
that for any ``possible" *m* term sequence *r*_{1},r_{2}, ...,r_{m}, there
are infinitely many primes *p* making it an *m* term permutation
chain of *k*th power residue modulo *p*, where *k* is an arbitrary
positive integer.
From our result,
we see that Vegh's theorem holds for any positive integer *k*, not
only for prime numbers. In fact, we prove our result in more
generality where the integer ring *\Z* is replaced by any
*S*-integer ring of global fields (i.e., algebraic number fields or
algebraic function fields over finite fields).

**2000 Mathematics Subject Classification.**
11A15, 11R04, 11R58.

**Key words and phrases.** *k*th power residue chain, global field, Chebotarev's density theorem.

**Full text (PDF)** (access from subscribing institutions only)
DOI: 10.3336/gm.46.1.03

**References:**

- H. Gupta,
*Chains of quadratic residues*, Math. Comp. **25** (1971), 379-382.

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- W. H. Mills,
*Characters with preassigned values*, Canad. J. Math. **15** (1963), 169-171.

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- J. Neukrich, Algebraic number theory, Springer-Verlag, Berlin, 1999.

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- M. Rosen, Number theory in function fields,
Springer-Verlag, New York, 2002.

MathSciNet

- E. Vegh,
*k*th-power residue chains, J. Number Theory **9** (1977), 179-181.

MathSciNet
CrossRef

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