Simply said, a field is a place where you can add, substract, multiply and divide, without leaving the set.
A field has several properties:
1. The rules of addition apply, and the field contains an additive identitive element
2. The rules of multiplication apply, and the field contains a multiplicative identity element
3. Every element in a field has an inverse
The set of integers is not a field, because integers don’t include fractions and so do not have multiplicative inverses.
The underlying set of a field determines whether a field is finite or infinite. If the set F is finite, then the field is said to be finite.
Infinite fields are not of particular interest in cryptographic applications, yet finite fields play a crucial role in many cryptographic algorithm.
Examples of infinite fields includes the real number, the rational numbers, the complex numbers and rational functions over a field.
The simplest finite field is modulo prime arithmetic.
Zp = {0, 1, …, p-1}, arithmetic mod p, where p is a prime, is a (finite) field
Examples:
Notice that Z4 (arithmetic mod 4) is not a field, since 2 has no inverse (look at the division table), there is no element x such that 2x = 1 (mod 4).
[Will post more on Finite Fields for Cryptographic Applications]
Apa ya syaratnya agar sebuah ring integer modulo n merupakan field?
jawabannya nanti di update posting ini ya
ini belum lengkap.
tanya2x terus aja biar tambah lengkap, hihihihi
I haven’t taken any relevant courses yet, so pardon the stupid question: In what sense are the rational numbers or the real numbers finite, or did you mean to say that they are infinite fields? If they are indeed finite, what would be an example of an infinite one?
@tommi: thx for visiting and asking such an interesting questions
yes! thx for pointing me the error. i was going to imply that rational and real numbers are indeed INFINITE fields [correction is going to be made after this]!!!
thx again, tommi!
Sorry yah……iseng menanggapi comment no 1, sambil ngetes latex.
Akan dibuktikan bahwa
adalah field jikka (iff) 
adalah bilangan prima. Misalkan 
adalah field. Jika $n$ tidak prima maka kita bisa tulis 
dengan $\latex 1<a,b<n$. Akibatnya $latex[a][b]=[ab]=0\in \mathbb{Z}_n$. It follows (upon multiplying by the inverse of a, ![[b]=0](http://l.wordpress.com/latex.php?latex=%5Bb%5D%3D0&bg=333333&fg=eeeeee&s=0 "[b]=0")
. So 
is a multiple of n (contradiction!).
then a=1 or 
. In the first case,then a has an inverse in Z_n and in the later case we have 
such that $ax+by=1$, which implies that x is the multiplicative inverse of a in Z_n. So Z_n is a field.
Coversely, suppose $n$ is prime. If
@5 hello temennya intan, makasih banyak ya udah ngelengkapin keterangannya. HARUS mampir dan iseng comment lagi ya!!!
btw satu pertanyaan penting: ITU NEMPELIN FORMULA LATEX KE COMMENT GIMANA CARANYA YA? seneng deh liatnya rapi…
thx!
Sorry, latexnya gak rapi. Gak biasa dengan penulisan syntax latex di wordpress. Untuk menuliskan latex tulis $latex\sqrt{b^2-4ac}$ (harusnya ada space diantara latex dan math formula). Hasilnya akan seperti ini
. Jadi perbedaannya dengan syntax latex yang biasa hanya di ruas kiri yg biasanya hanya dollar saja ($) diganti dengan $latex.
ok, thx tipsnya, yang jadi masalah, saya ga hafal perintah2x latex, kebiasaan pake lyx, he he he. kayaknya musti dibiasain sekarang ya. atau pake tools kayak di sini http://www.codecogs.com/components/equationeditor/equationeditor.php