A set with addition and multiplication satisfying the field axioms, including commutativity, associativity, identities, additive inverses, nonzero multiplicative inverses, and distributivity.

How is subtraction defined in a field?

By adding an additive inverse: $a-b=a+(-b)$.

How is division by a nonzero element defined?

As multiplication by its inverse: $a/b=ab^{-1}$, where $b\ne0$.

Why is $\mathbb{Z}$ not a field?

Because some nonzero integers do not have multiplicative inverses in $\mathbb{Z}$; for example, $1/2$ is not an integer.

What is the product of two complex numbers?

Using $i^2=-1$, $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$.

Abstract algebraLesson 1 of 4 in the course Fields and Galois theory

Fields: Axioms, Finite Fields, Extensions, and Galois Theory

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

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Summary

A field is an algebraic system in which addition, subtraction, multiplication, and division by nonzero elements work consistently. Fields generalize familiar number systems and provide the scalars used throughout algebra and linear algebra.

Subtraction is defined by $a-b=a+(-b)$ , and division by a nonzero $b$ is defined by $a/b=ab^{-1}$ . Division by zero is not defined.

All flashcards

What is a field?: A set with addition and multiplication satisfying the field axioms, including commutativity, associativity, identities, additive inverses, nonzero multiplicative inverses, and distributivity.
How is subtraction defined in a field?: By adding an additive inverse: $a-b=a+(-b)$ .
How is division by a nonzero element defined?: As multiplication by its inverse: $a/b=ab^{-1}$ , where $b\ne0$ .
Why is $\mathbb{Z}$ not a field?: Because some nonzero integers do not have multiplicative inverses in $\mathbb{Z}$ ; for example, $1/2$ is not an integer.
What is the product of two complex numbers?: Using $i^2=-1$ , $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$ .

Quiz questions

Which field axiom guarantees that multiplication distributes across a sum?Answer: $a(b+c)=ab+ac$
For which elements of a field is a multiplicative inverse required?Answer: Every nonzero element
Why is division by zero excluded in a field?Answer: Zero has no multiplicative inverse
Why are the integers not a field under ordinary addition and multiplication?Answer: Some nonzero integers lack integer reciprocals
If $z=a+bi$ and $w=c+di$ , what is the real part of $zw$ ?Answer: $ac-bd$

Based on the Wikipedia article Field (mathematics), licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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