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In this paper, we study about trigonometry in finite field, we know that
*p* elements, where
*p* is a prime number if and only if
*p* = 8
*k* + 1 or
*p* = 8
*k* -1. Let
*F* and
* K* be two fields, we say that
*F* is an extension of
*K*, if
*K*
⊆
*F* or there exists a monomorphism f:
*K→*
*F*. Recall that
*F*[x] is the ring of polynomial over
*F*. If
(means that
*F* is an extension of
*K*), an element
is algebraic over
*K* if there exists
such that
*f*(u) = 0 (see [1]-[4]). The algebraic closure of
*K* in
*F* is
, which is the set of all algebraic elements in
*F* over
*K*.

In this paper, we study about trigonometry in finite field, we know that

Let F and K be two fields, we say that F is an extension of K if

Definition. Let p be a prime number,

Note that symbol “|” is divisor or divides such that

Remark. 1) Recall that θ is a primitive kth root of unity if

2) We can define

Theorem 1. If K is a field with 9 elements and if 𝔽 is a finite extension of K, then the mapping

Proof. It is obviously that λ is onto and one to one (see [

Theorem 2. Let θ be a primitive kth root of unity. Then

Proof: Assume

Conversely, let

Corollary 3. If p ≠ 2 and θ is a primitive kth root of unity, then

Remark. We observe that since membership of

Lemma 4. Let θ be a primitive kth root of unity in

taining the ring

Proof. The formal derivative

Remark. For the basic properties of valuation rings, the reader can consult. In particular, it is worth recalling that each valuation ring is integrally closed in its quotient field K, and so, if

Corollary 5. Let (q, 10) = 1. Then

Proof. Define

and so

Remark. If in corollary 5 we take

Corollary 6. Assume (2, q) = 1. Then

Proof. Let

As before letting

Corollary 7. Let (6, q) = 1. Then

Proof. Let

Remark. If n = 0 and q = p above we have

Corollary 8. Let (q, 34) = 1. Then

The Formula in corollary 8 is quite complicated and one is naturally interested to know whether already some subformula of this formula is an element of

Indeed set ^{th} root of unity in

Corollary 9. Suppose that (q, 34) = 1, Then

Remark ( [

more easily. Indeed, for example, from c_{1} and c_{4} in

Theorem 10. Suppose (34, q) = 1, Then

Proof. If

if either

and only if

We want to prove that

Corollary 11. Assume (34, q) = 1. If

Therefore the inclusion

has been dealt with is lemma 4 from now on we assume

Definition. Let

Theorem 12. Let θ be a primitive kth root of unity. Then

(i)

(ii) k has the form 8m + 4 and

(iii) k has the form 8m + 4 and

Proof. Assume

Case (ii), Let

Case (iii),

equivalent to k = 8m + 4 and

Corollary 13. For any k, either

Proof. As

We conclude that in the field of real numbers, trigonometric ratios are defined as defined in finite fields. As well as relations between trigonometric ratios hold in the field of real numbers, finite fields are also established under the circumstances.

Amiri Naser,Hasani Fysal, (2016) Some New Results about Trigonometry in Finite Fields. Advances in Pure Mathematics,06,493-497. doi: 10.4236/apm.2016.67035