# 請教"體"的一些性質

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Since E is a finite field, we know that E*=E\{0} is a cyclic group of order 81-1=80.

Therefore, for all elements x in E* we have x^80-1=0.

Thus, all the elements in E* have the form

where 0<=n<80.

Note that x^40-1 and x^16-1 are the factors of x^80-1.

Find all (monic) irreducible polynomials of degree 2 over Z3.

There are nine possible polynomials of degree 2:

x^2, x^2+1, x^2+2, x^2+x, x^2+x+1, x^2+x+2, x^2+2x, x^2+2x+1, x^2+2x+2.

Check the roots of above polynomials by using 0, 1, 2. You shall find that x^2+1, x^2+x+2, and x^2+2x+2 are irreducible over Z3.

Find the number of elements α∈E, such that Z3(α) is a degree 2 extension of Z3.

We want to find α∈E such that [Z3(α):Z]=2; that is, find an irreducible polynomial f(x) of degree 2 and the root of f(x).

For example, f(x)=x^2+1 is irreducible and i is one of the roots of f(x). Let α=i then we have [Z3(i):Z]=2.

Find the number of elements α∈E, such that Z3(α) is a degree 4 extension of Z3.

Consider ξ16=cos(2π/5)+isin(2π/5), then (ξ16)^5-1=0.

But (ξ16)^5-1=(ξ16-1)[(ξ16)^4+(ξ16)^3+(ξ16)^2+ξ16+1], and ξ16≠1.

Therefore we have ξ16 is a root of the polynomial g(x)=x^4+x^3+x^2+x+1.

From the factorization of x^80-1 over Z3, we know that

g(x)=x^4+x^3+x^2+x+1 is irreducible, hence g(x) is the minimal polynomial of ξ16.

Let α=ξ16=cos(2π/5)+isin(2π/5), then we have [Z3(α):Z3]=4.

Findthe number of elements with order 80 in E*.

If(n,80)=1, then the order of ξn is80.

Hence there are φ(80)=80*(1-1/2)*(1-1/5)=32elements of order 80 in E*, where φ(n) is Euler's function.

2015-01-04 23:17:22 補充：

Find the number of (monic) irreducible polynomials of degree 4 over Z3.

There are 3 irreducible polynomials of order 1: x, x+1, x+2

There are 3 irreducible polynomials of order 2: x^2+1, x^2+x+2, x^2+2x+2

There are 8 irreducible polynomials of order 3:

(1)x^3+2x+1

(2)x^3+2x+2

(3)x^3+x^2+2

2015-01-04 23:28:07 補充：

因為有字數上的限制，所以我把剩下的都放在以下的網頁裡

2015-01-04 23:28:51 補充：

有更簡潔的證明歡迎提供，謝謝

2015-01-09 16:37:32 補充：

Find the number of (monic) irreducible polynomials of degree 4 over Z3.

這一題可以不必大費周章去數有幾個irreducible polynomials of degree 4

下面的定理可以派上用場

Lemma : If q(x) in Zp[x] is irreducible of degree n, then q(x)|(x^m-x), where m=p^n

2015-01-09 16:40:43 補充：

Let p=3, n=4, then q(x)|(x^81-x)

Since q(x) is of degree 4, q(x)|(x^80-1)

Hence x^80-1 has all irreducible polynomials of degree 4 over Z3.

The number of q(x) is the number of the irreducible polynomials of degree 4 in the linear factorization of x^80-1 over Z3; namely, 18.

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