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Structure of a Finite Field

Recall that a field F has characteristic p if p is the smallest positive integer such that for every nonzero element \alpha in F, we have p \alpha = 0. If no such integer exists, then F has characteristic 0.

Structure of a Finite Field

Recall that a field \(F\) has characteristic \(p\) if \(p\) is the smallest positive integer such that for every nonzero element \(\alpha\) in \(F\), we have \(p \alpha = 0\). If no such integer exists, then \(F\) has characteristic \(0\). From we know that \(p\) must be prime. Suppose that \(F\) is a finite field with \(n\) elements. Then \(n \alpha = 0\) for all \(\alpha\) in \(F\). Consequently, the characteristic of \(F\) must be \(p\), where \(p\) is a prime dividing \(n\). This discussion is summarized in the following proposition.

Throughout this chapter we will assume that \(p\) is a prime number unless otherwise stated.

Proposition

If \(F\) is a finite field of characteristic \(p\), then the order of \(F\) is \(p^n\) for some \(n \in {\mathbb N}\).

Proof

Let \(\phi : {\mathbb Z} \rightarrow F\) be the ring homomorphism defined by \(\phi(n) = n \cdot 1\). Since the characteristic of \(F\) is \(p\), the kernel of \(\phi\) must be \(p {\mathbb Z}\) and the image of \(\phi\) must be a subfield of \(F\) isomorphic to \({\mathbb Z}_p\). We will denote this subfield by \(K\). Since \(F\) is a finite field, it must be a finite extension of \(K\) and, therefore, an algebraic extension of \(K\). Suppose that \([F : K] = n\) is the dimension of \(F\), where \(F\) is a \(K\) vector space. There must exist elements \(\alpha_1, \ldots, \alpha_n \in F\) such that any element \(\alpha\) in \(F\) can be written uniquely in the form \[\begin{aligned}\end{aligned}\], where the \(a_i\)'s are in \(K\). Since there are \(p\) elements in \(K\), there are \(p^n\) possible linear combinations of the \(\alpha_i\)'s. Therefore, the order of \(F\) must be \(p^n\).

Let \(F\) be a field. A polynomial \(f(x) \in F[x]\) of degree \(n\) is separable if it has \(n\) distinct roots in the splitting field of \(f(x)\); that is, \(f(x)\) is separable when it factors into distinct linear factors over the splitting field of \(f\). An extension \(E\) of \(F\) is a separable extension of \(F\) if every element in \(E\) is the root of a separable polynomial in \(F[x]\).

The unique finite field with \(p^n\) elements is called the Galois field of order \(p^n\). We will denote this field by \(\gf(p^n)\). \(\gf(p^n)\) Galois field of order \(p^n\)

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.

Manje Akukho kalkuli yokuxazulula le nto, kodwa amaphuzu ayo anga kalkuli. Zama enye ngezansi, noma ubhale yakho.

Qhubeka usebenza

I-akhawunti emahhala ifaka iziphawulo kulesifundo ngasinye, irekhodi lokuthi yini oqedile, izinkinga zakho ezixazululiwe endaweni eyodwa, nodokotela owaziyo ongabuza mayelana nalekhasi. I-matematics ivulekile kubo bonke, noma ngabe babhalise noma cha.

Bhala Ngena

Amaphawu asetshenziswa lapha

Chofoza noma yiluphi uphawu lwesibonakaliso ukuze uthole ukuchaza okuphelele, isithombe, kanye nokuthi iyiphi inhlamvu ekhona ekhuluma ngaso.

Imibuzo abantu bebuza

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Amangxenye wale khasi aguqulwe kusuka Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). I-condensed futhi iphinde icaciswe lapha; amaphutha awusizo.

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