DONG QUAN NGOC NGUYEN

REPRESENTATION OF UNITS IN CYCLOTOMIC FUNCTION FIELDS DONG QUAN NGOC NGUYEN Contents 1 Introduction 1 2 Some basic notions 3 21 The Galois group Gal(K /k) 3 22 Representation of integers in O, and the multiplicative group G 4 3 Representation of units 5 References 14 Abstract In this paper, we prove a refinement of Hilbert s Satz 90 for the -cyclotomic function fields, where is a monic prime in F q[t ] Our result can be viewed as a function field analogue of Newman s theorem which is a refinement of Hilbert s Satz 90 for Q(ζ p)/q 1 Introduction Let q be be a power of a prime p, and let F q denote the finite field with q elements Let A F q [T ] be the ring of polynomials in the variable T over F q, and let k F q (T ) be the quotient field of A Let k alg denote an algebraic closure of k Let τ be the mapping defined by τ(x) x q, and let k τ denote the twisted polynomial ring Let C : A k τ (a C a ) be the Carlitz module, namely, C is an F q -algebra homomorphism such that C T T + τ Let m be a polynomial of positive degree, and set Λ m {λ k alg C m (λ) 0} We define a primitive m-th root of C to be a root of the polynomial C m (x) A[x] that generates the A-module Λ m Throughout the paper, for each polynomial m, we fix a primitive m-th root of C, and denote it by λ m The m-th cyclotomic function field, denoted by K m, is defined by K m k(λ m ) k(λ m ) It is known (see Hayes [3], or Rosen [7]) that K m is a field extension of k of degree Φ(m), where Φ( ) is a function field analogue of the classical Euler φ-function (We will recall the definition of Φ( ) in Section 2) There are many strong analogies between the m-th cyclotomic function fields K m and the classical cyclotomic fields Q(ζ m ) (see Goss [2], Hayes [3], Rosen [7], Thakur [8], or Villa Salvador [9]) Here the former letter m denotes a polynomial of positive degree in A, and the latter letter m stands for a positive integer in Z In this paper, we are interested in studying new analogous phenomena between the cyclotomic function fields and the classical cyclotomic fields Newman [6] proved a refinement of Hilbert s Satz 90 for the extensions Q(ζ p )/Q, where p > 3 is a prime, and ζ p denotes the p-th root of unity Recall that for a unit ɛ of norm 1 in Z[ζ p ], Hilbert s Satz 90 (see Hilbert [4], or Lang [5]) tells us that one can write ɛ as a quotient of conjugate integers in Z[ζ p ], ie, there exists an element δ Z[ζ p ] and an element σ Gal(Q(ζ p )/Q) such that δ Newman (see [6, Corollary, p357]) gave a refinement σ(δ) of Hilbert s Satz 90 by proving a sufficient and necessary condition for which such an element δ Date: April 18, 2017 1

2 DONG QUAN NGOC NGUYEN can be chosen to be a unit in Z[ζ p ], and thus ɛ is a quotient of conjugate units in Z[ζ p ] In order to obtain this result, Newman proved a stronger result that provides a unique representation of a unit of norm 1 as a product of a power of a unit 1 ζe p 1 ζ p with a quotient of conjugate units More precisely, Newman proved the following Theorem 11 (Newman, see [6, Theorem, p353]) Fix an isomorphism between Gal(Q(ζ p /Q) and F p Z/(p 1)Z Let e be an integer such that e modulo p is a primitive root modulo p, and let σ e be the unique element of the Galois group Gal(Q(ζ p )/Q) that sends ζ p to ζp e Let G p be the multiplicative group consisting of all units of the form δ σ e (δ), where δ is a unit in Z[ζ p] Then for any unit ɛ of norm 1 in Z[ζ p ], there exist an integer l with 0 l p 2, and a unit α G p such that ( 1 ζ e ) l p α 1 ζ p ( 1 ζ e ) l p Furthermore the above representation is unique, ie, if α for some integer 1 ζ p l with 0 l p 2 and some unit α G p, then l l and α α Remark 12 In the original statement of Newman s theorem (see [6, Theorem, p353]), Newman assumes that ɛ is any unit in Q(ζ p ), which is erroneous Indeed ɛ must be a unit of norm 1 so that Hilbert s Satz 90 can apply In addition, the conclusion of Newman s theorem (see equation (1) in [6, Theorem, p353]) indicates that the unit ɛ must be of norm 1 (Note that in [6], Newman used the notation α instead of ɛ as in our paper) The aim of this paper is to prove a function field analogue of the above theorem, which can be viewed as a refinement of Hilbert s Satz 90 for the extensions K /k More explicitly, our main goal in this paper is to prove the following Theorem 13 (See Theorem 37) Let be a monic prime in A Let K be the -th cyclotomic function field, and let O be the ring of integers of K Let g be a primitive root modulo, and let σ g be the unique element of the Galois group Gal(K /k) that sends to C g ( ) Let G be the multiplicative group consisting of all units of the form δ σ g (δ), where δ is a unit in O Then for any unit ɛ of norm 1 in O, there exist an integer l with 0 l q deg( ) 2, and a unit α G such that ( ) l λ α C g ( ) ( ) l λ Furthermore the above representation is unique, ie, if α for some integer C g ( ) l with 0 l q deg( ) 2 and some unit α G, then l l and α α One can replace 1 ζe p in Theorem 11 by its inverse 1 ζ p 1 ζ p 1 ζp e, and obtain a similar representation for units of norm 1 that is equivalent to that of Theorem 11 It is well-known (see Rosen [7, Proposition 126]) that C g ( ) is a unit in O, and is a function field analogue of the unit 1 ζ p 1 ζp e in the number field setting Hence Theorem 13 can be viewed as a function field analogue of Newman s theorem

REPRESENTATION OF UNITS IN CYCLOTOMIC FUNCTION FIELDS 3 Using the same ideas as in Newman [6], we also obtain a sufficient and necessary condition for which a unit of norm 1 in O can be written as a quotient of conjugate units This is a refinement of Hilbert s Satz 90 for the extensions K /k The structure of the paper is as follows In Section 2, we recall some basic notions and notation that will be used throughout the paper In Section 3, we prove Theorem 13 (see Theorem 37), and consequently obtain a refinement of Hilbert s Satz 90 (see Corollary 39) The proof of Theorem 13 uses the same ideas and approach as in the proof of Newman s theorem, but we need to introduce some modifications in many places to adapt the proof of Newman s theorem into the function field setting 2 Some basic notions The aim of this section is to recall some basic notions and fix some notation that that will be used throughout the paper For a polynomial m A of positive degree, we define Φ(m) to be the number of nonzero polynomials of degree less than deg(m) and relatively prime to m, ie, Φ(m) card((a/ma) ) The function Φ( ) is a function field analogue of the classical Euler φ-function Let m α s1 1 s h h be the prime factorization of m, where α F q, the i are monic primes in A, and the s i are positive integers It is well-known (see [7, Proposition 17]) that Φ(m) h h Φ( si i ) i1 i1 (q deg( s i i ) q deg( s i 1 i ) ) In particular, when m s for some monic prime and some positive integer s, Φ( s ) q deg( s) q deg( s 1) Recall that the m-th cyclotomic polynomial, denoted by Ψ m (x), is the minimal polynomial of λ m over k It is well-known (see Hayes [3], or Rosen [7]) that Ψ m (x) is the monic irreducible polynomial of degree Φ(m) with coefficients in A such that Ψ m (λ m ) 0 When m s for some monic prime and some positive integer s, we know from [3, Proposition 24] that (1) Ψ s(x) C s(x)/c s 1(x) From Rosen [7, Proposition 1211], one can write C (x) x + [, 1]x q +, +[, deg( ) 1]x qdeg( ) 1 + x qdeg( ), where [, i] A for each 1 i deg( ) 1 For s 1, the equation (1) tells us that Ψ (x) C (x) (2) + [, 1]x q 1 +, +[, deg( ) 1]x qdeg( ) 1 1 + x qdeg( ) 1 x When x 0, we obtain the following elementary result that will be useful in the proof of our main theorem Proposition 21 Ψ (0) 21 The Galois group Gal(K /k) For the rest of this paper, fix a monic prime of positive degree Let K be the -th cyclotomic function field, and let O be the ring of integers of K Note that when q 2 and deg( ) 1, K k So without loss of generality, we further assume that ( ) q > 2 or deg( ) > 1

4 DONG QUAN NGOC NGUYEN There is an isomorphism between Gal(K /k) and the multiplicative group (A/ A) (see Rosen [7, Chapter 12]) For each element m (A/ A), there exists an isomorphism σ m Gal(K /k) that is uniquely determined by the relation σ m ( ) C m ( ) The correspondence σ m m is an isomorphism from Gal(K /k) into the group (A/ A) Throughout the paper, fix an element g A such that g is a primitive root modulo, ie, g is a generator of the group (A/ A) Note that the order of (A/ A) is Φ( ) q deg( ) 1, and thus one can write (A/ A) g {g e 0 e q deg( ) 2} The next result is well-known Proposition 22 The isomorphism σ g Gal(K /k) is a generator of the cyclic group Gal(K /k) More precisely, σ g e( ) σg(λ e ) for each 0 e q deg( ) 2 Throughout the paper, we denote by Norm K /k( ) the norm of K over k Proposition 22 implies that for each element α K, Norm K /k(α) σ Gal(K /k) σ(α) q deg( ) 2 e0 σ e g(α) The following elementary result will be useful in the proof of our main theorem Proposition 23 Let α be a unit in O of norm 1 Then there exists an element δ O such that α δ σ g (δ) Proof Applying Hilbert s Satz 90 (see Lang [5]), there exists an element γ K such that (3) α γ σ g (γ) Since K k( ), one can write i γ a iλ i, b where the a i are in A, and b A \ {0} From (3), one gets i a iλ i α b ( i σ a ) δ iλ i σ g (δ), g b where δ i a iλ i A[ ] Since O A[ ] (see Rosen [7, Proposition 129]), our contention follows 22 Representation of integers in O, and the multiplicative group G Since the degree of K over k is Φ( ) q deg( ) 1, the next result is obvious Proposition 24 For each integer α O, there exists a unique polynomial P α (x) A[x] of degree at most q deg( ) 2 such that α P α ( ) Furthermore α 0 if and only if the polynomial P α (x) is identically zero For the rest of the paper, for each integer α O, we always denote by P α the unique polynomial satisfying α P α ( ) in Proposition 24 We call P α the polynomial representing α

REPRESENTATION OF UNITS IN CYCLOTOMIC FUNCTION FIELDS 5 We now introduce the multiplicative group G that will be of interest in this paper Let G be the set consisting of all elements of the form α σ g (α), where α is a unit in O It is clear that G is a group Throughout this paper, we denote by U (O ) be the group of units of norm 1 in O The following result follows immediately Proposition 25 (i) G is a subgroup of U (O ) (ii) { } α G σ g (α) α is a unit in O (iii) { } Pα ( ) G P α (C g ( )) α is a unit in O, and P α (x) A[x] is the polynomial representing α 3 Representation of units In this section, we will prove Theorem 13 (see Theorem 37 As a consequence, we obtain a refinement of Hilbert s Satz 90 (see Corollary 39) We begin by proving several lemmas that we will need in the proof of our main theorem The next result is a function field analogue of Newman [6, Lemma 1] We follow the same ideas as in the proof of Newman [6, Lemma 1] to prove the next lemma Lemma 31 Let α be an integer in O, and let P α (x) A[x] be the polynomial representing α Assume that the following are true (i) P α(c g ( )) is a unit in O (Recall that g is a generator of the group (A/ A) ) P α ( ) (ii) gcd(p α (0), ) 1 (iii) the content of the polynomial P α (x) is 1, ie, the greatest common divisor of all coefficients of P α (x) is 1 Then α is a unit in O Remark 32 By (ii) in Lemma 31, one sees that the polynomial P α (x) is nonzero, and it thus follows from Proposition 24 that α P α ( ) 0 Proof Assume the contrary, ie, α is a non-unit in O Since α P α ( ), P α ( ) is also a non-unit in O Hence there exists a prime ideal q in O that divides P α ( ) Thus the ideal σ g (q) is prime, and is a prime ideal divisor of σ g (P α ( )) Since σ g (P α ( )) P α (σ g ( )) P α (C g ( )), it follows that σ g (q) is a prime ideal divisor of P α (C g ( )) We deduce from (i) that σ g (q) is also a prime ideal divisor of P α ( ) Since σ g is a generator of the Galois group Gal(K /k), every conjugate of the prime ideal q is a prime ideal divisor of P α ( ), ie, for every element σ Gal(K /k), σ(q) is a prime ideal divisor of P α ( ) We contend that gcd(p α ( ), ) 1; otherwise since O is a prime ideal in O (see Rosen [7, Proposition 127]), we deduce that O divides P α ( ), and thus P α (0) P α ( ) 0 (mod ) Since O ( O ) qdeg( ) 1 (see Rosen [7, Proposition 127]), we deduce that P α (0) 0 (mod ), which is a contradiction to (ii)

6 DONG QUAN NGOC NGUYEN Since the prime ideal σ(q) divides P α ( ) for every σ Gal(K /k), we deduce that σ(q) O for every σ Gal(K /k) Since q is a prime ideal, we know that Norm K /k(q) (pa) r for some monic prime p A and some positive integer r By [1, Theorem 351], and since O ( O ) qdeg( ) 1, we deduce that p By Rosen [7, Proposition 127], K is unramified at p, and thus po is the product of the distinct conjugates of q Therefore po divides P α ( )O, and thus (4) (5) One can write P α ( ) p P α (x) O h ɛ i x i, where the ɛ i are in A with ɛ h 0, and h deg(p α (x)) q deg( ) 2 By [7, Proposition 129], O A[ ], and since the minimal polynomial of over k is the -th cyclotomic polynomial Ψ (x) A[x] of degree exactly q deg( ) 1 (see Hayes [3] or Section 2), it follows from (4) and (5) that there exists an element of the form q deg( ) 2 j0 κ j λ j O A[ ] with the κ j A such that (6) Therefore P α ( ) p h i0 h i0 i0 q ɛ deg( ) i 2 p λi κ j λ j j0 ( ) q deg( ) ɛi p κ 2 i λ i κ i λ i 0 ih+1 Since h q deg( ) 2 and the minimal polynomial of over k is the -th cyclotomic polynomial Ψ (x) A[x] of degree exactly q deg( ) 1, we deduce from (6) that ɛ i p κ i 0 for every 0 i h Thus ɛ i pκ i for every 0 i h Therefore p divides gcd 0 i h (ɛ i ) This implies that the content of P α (x) is divisible by p, which is a contradiction to (iii) in Lemma 31 Thus α is a unit in O (7) (8) For each e 1, set One can also write (7) in the form ( ) ρ e σe 1 g ( ) σg(λ e σg e 1 λ ) σ g ( ) ρ e C g e 1() C g e( ) Let P ρe (x) A[x] be the polynomial representing ρ e From the above equation, one gets ( ) σ e 1 g ( ) P ρe (C g ( )) P ρe (σ g ( )) σ g (P ρe ( )) σ g (ρ e ) σ g σg(λ e, )

REPRESENTATION OF UNITS IN CYCLOTOMIC FUNCTION FIELDS 7 and thus (9) P ρe (C g ( )) σe g( ) σ e+1 g ( ) ρ e+1 P ρe+1 ( ) Lemma 33 Let l be an integer 2, and set l h ρ e l Λ ρ e 1 Then (10) h2 e2 Proof For each h 1, we see that and it thus follows from (8) that l ρ h Λ 1 ρ 1 ρ l 1 1 h2 Therefore (10) follows immediately h h2 e2 ( ) l C g l( ) Λ λ C g ( ) l h2 h e2 ρ e ρ e 1 ρ h ρ 1, C g h 1( ) C g h( ) P ρe 1 (C g ( )) P ρe 1 ( ) ( Cg ( ) ) l 1 C g ( ) C g l( ) Let l q deg( ) 1 Note that ( ) in Subsection 21 implies that l 2 Since we deduce from Lemma 33 that (11) C g q deg( ) 1( ) σ g q deg( ) 1( ) σ qdeg( ) 1 g ( ), Λ ( ) 1 q deg( ) λ C g ( ) We summarize the above discussion in the following result Corollary 34 Let l be an integer Then there exist an integer e with 0 e q deg( ) 2 and an integer h such that the following are true: (i) l (q deg( ) 1)h + e; and (ii) ( ) l ( ) e λ λ Λ h C g ( ) C g ( ) From the definition of Λ in Lemma 33, Λ 1 clearly belongs to the multiplicative group G, and so does Λ h for any integer h (Recall that G is defined in Subsection 22) From Proposition 25 and Corollary 34, we obtain the following result Corollary 35 Let l be an integer Then there exist an integer e with 0 e q deg( ) 2 Q( ) and an element Q(C g ( )) G such that ( ) l ( ) e λ λ Q( ) C g ( ) C g ( ) Q(C g ( )) Lemma 36 Let Q(x) be a polynomial in A[x] such that Q( ) is a unit in O gcd(q(0), ) 1 Then

8 DONG QUAN NGOC NGUYEN Proof Let Q( ) By assumption, ɛ is a unit in O Assume the contrary, ie, divides Q(0) It is known (see Rosen [7, Proposition 127]) that O ( O ) qdeg( ) 1 Thus divides Q(0), and hence Q( ) Q(0) 0 (mod ), which is a contradiction since ɛ is a unit in O and O is a prime ideal in O (see Rosen [7, Proposition 127]) The next result is our main theorem in this paper, which can be viewed as a function field analogue of Newman [6, Theorem] The proof of the next theorem follows the same ideas as in the proof of Newman s theorem, but we need some modifications to adapt the proof of Newman s theorem into the function field setting Theorem 37 Let ɛ be a unit in O of norm 1 Then there exist an integer l with 0 l q deg( ) 2, and a polynomial Q(x) A[x] of degree at most q deg( ) 2 with Q( ) being a unit in O such that the following are true: (R1) ɛ can be represented in the form ( ) l ( ) λ Q(λ ) C g ( ) Q(C g ( )) (R2) The representation of ɛ in (R1) is unique, except that Q(x) can be replaced by υq(x) for some unit υ F q ; more precisely, if there exist an integer l with 0 l q deg( ) 2, and a polynomial Q (x) A[x] of degree at most q deg( ) 2 with Q ( ) being a unit in O such that ( λ C g ( ) ) l ( ) Q ( ), Q (C g ( )) then l l and Q (x) υq(x) for some unit υ F q Proof Let P ɛ (x) A[x] be the polynomial representing ɛ, ie, P ɛ ( ) By Lemma 36, (12) gcd(p ɛ (0), ) 1 By (12) and since g is a generator of the cyclic group (A/ A), and #(A/ A) q deg( ) 1, there exists an integer h with 0 h q deg( ) 2 such that (13) (14) Set P ɛ (0) g h (mod ) ( ) h 1 Cg ( ) α for some α K Since C g( ) is a unit in O, we see that α is a unit in O Let P α (x) A[x] be the polynomial representing α, and set ( ) q Cg (x) deg( ) +h 2 (15) Q(x) P α (x) x From Rosen [7, Proposition 1211], one can write C g (x) A[x] in the form C g (x) gx + [g, 1]x q + + [g, deg(g) 1]x qdeg(g) 1 + [g, deg(g)]x qdeg(g),

REPRESENTATION OF UNITS IN CYCLOTOMIC FUNCTION FIELDS 9 where [g, i] A for each 1 i deg(g) Thus C g (x) (16) g + [g, 1]x q 1 + + [g, deg(g)]x qdeg(g) 1 A[x] x Since q deg( ) + h 2 > 0, we see that Q(x) A[x] We see from (14) that ( ) q Cg ( ) deg( ) +h 2 Q( ) P α ( ) ( ) q Cg ( ) deg( ) 1 ɛ, and thus ( ) q Cg ( ) deg( ) 1 Q( ) P ɛ ( ) Q( ) ( Cg ( ) ) q deg( ) 1 0 ( ) q Cg (x) deg( ) 1 Since Q(x) P ɛ (x) A[x], and the -th cyclotomic polynomial Ψ (x) x is the minimal polynomial of, it follows from the above equation that Ψ (x) divides ( ) q Cg (x) deg( ) 1 Q(x) P ɛ (x) in A[x] Thus there exists a polynomial, say T (x) A[x] x such that ( ) q Cg (x) deg( ) 1 (17) Q(x) P ɛ (x) T (x)ψ (x) x Setting Z(x) C g(x) A[x], we see from (16) that x (18) Z(0) g From Proposition 21, and (17), we deduce that Q(0) Z(0) qdeg( ) 1 P ɛ (0) T (0)Ψ (0) T (0) 0 From (13), (15), (18), and the above equation, we deduce that (mod ) g qdeg( ) +h 2 P α (0) Z(0) qdeg( ) +h 2 P α (0) Q(0) Z(0) qdeg( ) 1 P ɛ (0) g qdeg( ) +h 1 and thus (19) P α (0) g (mod ) We know that σ g ( ) C g ( ), and thus C g( ) σ g( ) Since σ g is a generator of the Galois group Gal(K /k), it follows that ( ) ( ) Cg ( ) σg ( ) Norm K /k 1 Norm K /k Since ɛ is of norm 1, equation (14) and the above equation imply that α is also of norm 1 Applying Proposition 23, there exists an element δ O such that (20) (21) α δ σ g (δ) Let P δ (x) A[x] be the polynomial representing δ Since δ P δ ( ), one can write P δ ( ) δ λ e η, (mod ),

10 DONG QUAN NGOC NGUYEN where η O such that gcd(, η) 1, and e is a nonnegative integer Let P η (x) A[x] be the polynomial representing η Since η P η ( ) P η (0) (mod ), and O ( O ) qdeg( ) 1, we see that if divides P η (0), then the prime ideal O divides η, which is a contradiction Hence (22) gcd(p η (0), ) 1 Note that the polynomial P η (x) is not identical to zero since η 0 Let c(p η ) A be the content of the polynomial P η (x), ie, the greatest common divisor of all nonzero coefficients of P η (x) in A Obviously c(p η ) 0 Set (23) and let (24) R(x) P η(x) c(p η ) A[x], β R( ) P η( ) c(p η ) η c(p η ) O A[ ] Let P β (x) A[x] be the polynomial representing β Since deg(r(x)) deg(p η (x)) q deg( ) 2, the uniqueness implies that P β (x) R(x) (see Proposition 24) It follows from (23) that (25) α λe P η ( ) σ g (λ e P η ( )) λ e P η ( ) σ g (λ e )σ g (P η ( )) P β (x) P η(x) c(p η ) From (20) and (21), and since c(p η ) A, one sees that ( λ σ g ( ) and thus α It therefore follows from (14) that ) e c(p η )P β ( ) σ g (c(p η )P β ( )) ( ) e ( ) e λ P β ( ) C g ( ) P β (σ g ( )) λ P β ( ) C g ( ) P β (C g ( )) ( ) e λ P β ( ) C g ( ) σ g (P β ( )), (26) ( ) e h+1 λ P β ( ) C g ( ) P β (C g ( )) Since (i) C g ( ) is a unit in O, we deduce that P β ( ) P β (C g ( )) is a unit in O, or equivalently P β(c g ( )) P β ( ) is a unit in O From (22), (25), and since c(p η ) is the content of P η (x), we deduce that (ii) gcd(p β (0), ) 1 (iii) the content of the polynomial P β (x) is 1 Since P β (x) satisfies all the conditions in Lemma 31, we deduce that β P β ( ) is a unit P β ( ) in O, and thus P β (C g ( )) belongs to the group G (Recall that G is defined in Subsection

REPRESENTATION OF UNITS IN CYCLOTOMIC FUNCTION FIELDS 11 22) By Corollary 35, there exist an integer l with 0 l q deg( ) 2, and an element P( ) P(C g ( )) G such that ( ) e h+1 ( ) l λ λ P( ) C g ( ) C g ( ) P(C g ( )), and it thus follows from (26) that ( ) l ( ) λ P(λ ) P β ( ) (27) C g ( ) P(C g ( )) P β (C g ( )) Since G is a multiplicative group, it follows from Proposition 25(iii) that there exists a polynomial Q(x) A[x] of degree at most q deg( ) 2 such that Q( ) is a unit in O, and Hence we deduce from (27) that P( ) P β ( ) P(C g ( )) P β (C g ( )) Q() Q(C g ( )) ( ) l ( ) λ Q(λ ), C g ( ) Q(C g ( )) which proves the first part of the theorem We now prove the uniqueness Assume that there exist another integer l with 0 l q deg( ) 2, and another polynomial Q (x) A[x] of degree at most q deg( ) 2 with Q ( ) being a unit in O such that (28) Hence (29) ( λ C g ( ) ( Cg ( ) ) l ( Q(λ ) Q(C g ( )) ) ) l ( ) Q(Cg ( )) Q( ) ( ) l ( ) λ Q ( ) C g ( ) Q (C g ( )) ( Cg ( ) ) l ( ) Q (C g ( )) Q ( ) By Rosen [7, Proposition 1211], C g (x) A[x] can be written in the form C g (x) gx + [g, 1]x q + + [g, deg(g)]x qdeg(g), where the [g, i] are in A, and [g, deg(g)] F q (30) and (31) C g (0) 0, C g ( ) g (mod ) is the leading coefficient of g It follows that Furthermore we deduce from Lemma 36 that gcd(q(0), ) 1 and gcd(q (0), ) 1 In particular this implies that Q(0), Q (0) are nonzero elements in A Hence it follows from (29), (30), and (31) that g l g l (mod ) Since O ( O ) qdeg( ) 1 (see Rosen [7, Proposition 127]), we deduce that g l g l (mod ) Without loss of generality, we assume that l l If l > l, then g l l 1 (mod ),

12 DONG QUAN NGOC NGUYEN and since g is a primitive root modulo, the above equation implies that #(A/ A) q deg( ) 1 divides l l This is a contradiction since 0 < l l < q deg( ) 1 Hence (32) l l By (28), and since σ g ( ) C g ( ), we deduce that Q( ) Q ( ) Q(C g( )) Q (C g ( )) σ g ( Q(λ ) Q ( ) Thus the unit Q() Q ( ) in O is invariant under the action of σ g Since σ g is a generator of the cyclic group Gal(K /k), the unit Q() is also invariant under the action of each member of Q ( ) Gal(K /k), and thus Q() Q ( ) is a unit in A This implies that Q() Q ( ) F q, and therefore there exists a unit υ F q such that (33) Q ( ) υq( ) Set κ Q ( ) By the assumption, we know that κ is a unit in O Since the polynomial Q (x) is of degree at most q deg( ) 2, it follows from Proposition 24 that Q (x) P κ (x), where P κ (x) A[x] is the polynomial representing κ On the other hand, we know from (33) that κ υq( ) Note that υq(x) is a polynomial in A[x] of degree at most q deg( ) 2 since υ F q Hence repeating the same arguments as above, one sees that υq(x) P κ (x), and thus (34) Q (x) υq(x) The second part of the theorem follows immediately from (32) and (34) Theorem 37 can be interpreted in terms of group theory Corollary 38 (i) G is a subgroup of U (O ) such that [U (O ) : G ] q deg( ) 1 (Recall that U (O ) is the group of units of norm 1 in O ) (ii) The quotient group U (O )/G is cyclic of order q deg( ) 1; furthermore U (O )/G is generated by C g ( ) G Hilbert s Satz 90 (see Hilbert [4]) tells us that for a unit ɛ U (O ), there exists an element δ O such that δ (see Proposition 23) The next result is a refinement of Hilbert s σ g (δ) Satz 90 for the extension K /k, which gives a sufficient and necessary condition under which an element in O is a quotient of conjugate units Corollary 39 Let ɛ U (O ) Then ɛ is the quotient of conjugate units in O if and only if P ɛ (0) 1 (mod ), where P ɛ (x) A[x] is the polynomial representing ɛ Proof Assume that ɛ is the quotient of conjugate units in O, ie, δ for some unit σ g e(δ) δ O, and some integer e with 0 e q deg( ) 2 (Recall that σ g is a generator of the Galois group Gal(K /k)) If e 0, then 1, and it is easy to see that the polynomial P ɛ (x) P 1 (x) 1 Hence P 1 (0) 1 (mod ) )

REPRESENTATION OF UNITS IN CYCLOTOMIC FUNCTION FIELDS 13 We now consider the case where e 1 Set e 1 α σ g i(δ) δ σ g (δ) σ g e 1(δ) i0 One can immediately verify that α is a unit in O, and (35) By Proposition 25, (35) that (36) δ σ g e(δ) α σ g (α) α σ g (α) belongs to the group G Since P ɛ ( ), we deduce from P ɛ ( ) α σ g (α) P α( ) P α (C g ( )), where P α (x) is the polynomial representing α Following the same arguments as in the proof of Theorem 37 (see equation (30)), we deduce that C g (0) 0 Using Lemma 36, we know that gcd(p α (0), ) 1; hence gcd(p α (0), ) 1 since O ( O ) qdeg( ) 1 and O is a prime ideal in O (see Rosen [7, Proposition 127]) In particular, this implies that P α (0) 0 (mod ) By (36), we see that and thus (37) P ɛ (0) P ɛ ( ) P α( ) P α (C g ( )) P α(0) P α (C g (0)) P α(0) P α (0) 1 P ɛ (0) 1 Conversely assume that ɛ U (O ) such that P ɛ (0) 1 (mod ) (mod ) By Theorem 37, one can write ɛ in the form ( ) l (38) λ Q( ) P ɛ ( ) C g ( ) Q(C g ( )), (mod ), where l is an integer such that 0 l q deg( ) 2, and Q(x) is a polynomial in A[x] of degree at most q deg( ) 2 such that Q( ) is a unit in O One can write (38) in the form ( ) l (39) Cg ( ) P ɛ ( ) Q() Q(C g ( )) Following the same arguments as in the proof of Theorem 37 (see equation (31)), we know that C g ( ) g (mod ), and thus (40) ( ) l Cg ( ) g l (mod ) Repeating the same arguments as in the first part of this proof, one can show that Q( ) Q(C g ( )) Q(0) Q(C g (0)) Q(0) Q(0) 1 and it thus follows from (39) and (40) that (41) g l P ɛ (0) 1 (mod ) (mod ),

14 DONG QUAN NGOC NGUYEN Since g l P ɛ (0) A, and O ( O ) qdeg( ) 1 (see Rosen [7, Proposition 127]), we deduce from (41) that and it thus follows from (37) that g l P ɛ (0) 1 g l 1 (mod ), (mod ) Since g is a generator of the cyclic group (A/ A), and the order of (A/ A) is q deg( ) 1, the above equation implies that l 0 Therefore we deduce from (38) that Q() Q(C g ( )) Q() Q(σ g ( )) Q() σ g (Q( )) κ σ g (κ), where κ Q( ) is a unit in O Thus our contention follows Remark 310 Corollary 39 is a function field analogue of Newman [6, Corollary, p357] References [1] DM Goldschmidt, Algebraic functions and projective curves, Graduate Texts in Mathematics, 215 Springer Verlag, New York, (2003) [2] D Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 35, Springer-Verlag, Berlin, (1996) [3] DR Hayes, Explicit class field theory for rational function fields, Trans Amer Math Soc 189 (1974), 77 91 [4] D Hilbert, Die Theorie der algebraischen Zahlköper, Jahresber Deutsch Math-Verein 4 (1897), 175 546 Reprinted in the first volume of Hilbert s collected works, Springer (1932) [5] S Lang, Algebra, Addison Wesley Publishing Co, Inc, Reading, Mass (1965) [6] M Newman, Cyclotomic units and Hilbert s Satz 90, Acta Arith 41 (1982), no 4, 353 357 [7] M Rosen, Number theory in function fields, Graduate Texts in Mathematics, 210 Springer-Verlag, New York (2002) [8] DS Thakur, Function Field Arithmetic, World Scientific Publishing Co, Inc, River Edge, NJ (2004) [9] GD Villa Salvador, Topics in the theory of algebraic function fields, Mathematics: Theory & Applications Birkhäuser Boston, Inc, Boston, MA (2006) Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana 46556, USA E-mail address: dongquanngocnguyen@ndedu