Field Theory Calculator

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Example

Created on 2024-06-20Asked by Scarlett Scott (Solvelet student)

Define the concept of a field in abstract algebra.

Solution

In abstract algebra, a field is a set equipped with two binary operations, addition and multiplication, which satisfy the following properties: 1. Closure: For any two elements

a

and

b

in the field,

a + b

and

a \times b

are also in the field. 2. Associativity: Addition and multiplication are associative operations. That is, for any three elements

a

b

, and

c

in the field,

a + (b + c) = (a + b) + c

and

a \times (b \times c) = (a \times b) \times c

. 3. Identity Elements: There exist additive and multiplicative identity elements denoted by

0

and

1

, respectively, such that for any element

a

in the field,

a + 0 = a

and

a \times 1 = a

. 4. Additive Inverses: For every element

a

in the field, there exists an additive inverse denoted by

-a

, such that

a + (-a) = 0

. 5. Multiplicative Non-Zero Elements: Every non-zero element in the field has a multiplicative inverse. That is, for every non-zero element

a

in the field, there exists an element

b

such that

a \times b = 1

. 6. Distributive Property: Multiplication distributes over addition. That is, for any three elements

a

b

, and

c

in the field,

a \times (b + c) = (a \times b) + (a \times c)

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Lucas Martin on Solvelet

1. Verify whether the set of rational numbers

\mathbb{Q}

forms a field under addition and multiplication.2. Determine whether the set of real numbers

\mathbb{R}

is a field under the usual operations of addition and multiplication.

DefinitionIn mathematics, field theory is a branch of abstract algebra which studies the properties of fields: sets endowed with two distinct operations of addition and multiplication, with more characteristic axioms (for instance, some fields having characteristic zero, some fields are locally compact, etc.). Field theory provides tools like fields, extensions of fields, algebraic extensions, Galois theory etc. that help to study properties of equations at the level of polynomial roots and other algebraic structures. Field theory is also used to study number theory, discussion of algebraic geometry, cryptography and theoretical physics. For example, the set of rational numbers Q forms a field under addition and multiplication, where every nonzero element has a multiplicative inverse.

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