What is a finite field?

A field containing finitely many elements, with division defined for every nonzero element.

What is a primitive element of GF(q)?

A generator of the multiplicative group of all nonzero elements of GF(q).

What is the Frobenius automorphism of GF($p^n$)?

The map $x\mapsto x^p$, which fixes the prime subfield GF(p).

What is a discrete logarithm?

For primitive element $a$, the exponent $n$ such that $x=a^n$.

Why does an irreducible polynomial produce a field quotient?

Every nonzero equivalence class in the quotient has a multiplicative inverse.

Abstract algebraLesson 1 of 1 in the course Fields and Galois theory

Finite Fields: Structure, Construction, and Multiplicative Groups

5 flashcards · 5 quiz questions · 5 match pairs · 4 fill-in sentences · English

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Summary

This lesson introduces finite fields and the rules that govern their size and arithmetic. These ideas are the foundation for understanding how finite fields are built and used.

A field is a set with addition, subtraction, multiplication, and division by nonzero elements, satisfying familiar arithmetic laws such as associativity, distributivity, and the existence of additive and multiplicative identities. A finite field is a field with only finitely many elements. It is also called a Galois field, often written GF(q) or F_q.

All flashcards

What is a finite field?: A field containing finitely many elements, with division defined for every nonzero element.
What is a primitive element of GF(q)?: A generator of the multiplicative group of all nonzero elements of GF(q).
What is the Frobenius automorphism of GF( $p^n$ )?: The map $x\mapsto x^p$ , which fixes the prime subfield GF(p).
What is a discrete logarithm?: For primitive element $a$ , the exponent $n$ such that $x=a^n$ .
Why does an irreducible polynomial produce a field quotient?: Every nonzero equivalence class in the quotient has a multiplicative inverse.

Quiz questions

What does the classification of finite fields guarantee about two fields with the same finite order?Answer: They are isomorphic, so a bijection preserves addition and multiplication.
When does GF( $p^m$ ) occur as a subfield of GF( $p^n$ )?Answer: Exactly when $m$ divides $n$ .
In the quotient GF(p)[X]/(P(X)), what does irreducibility of $P(X)$ ensure?Answer: Every nonzero class has a multiplicative inverse.
When does GF(q) contain a primitive $n$ th root of unity?Answer: When $n$ divides $q-1$ .
How many primitive elements does GF(16) have?Answer: 8.

Based on the Wikipedia article Finite field, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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