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Fields

Description: This quiz will test your knowledge of fields, which are algebraic structures that generalize the concept of number systems.
Number of Questions: 15
Created by:
Tags: algebra fields mathematics
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Which of the following is a field?

  1. The set of integers

  2. The set of rational numbers

  3. The set of real numbers

  4. The set of complex numbers

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Correct Option: D
Explanation:

A field is a set of elements with two operations, addition and multiplication, that satisfy certain properties. The set of complex numbers is a field because it satisfies these properties.

What is the multiplicative identity of a field?

  1. 0

  2. 1

  3. -1

  4. 2

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Correct Option: B
Explanation:

The multiplicative identity of a field is the element that, when multiplied by any other element of the field, results in that element. In the set of real numbers, the multiplicative identity is 1.

What is the additive inverse of an element in a field?

  1. The element itself

  2. The opposite of the element

  3. The reciprocal of the element

  4. The square root of the element

Show answer
Correct Option: B
Explanation:

The additive inverse of an element in a field is the element that, when added to the original element, results in the additive identity. In the set of real numbers, the additive inverse of an element is its opposite.

What is the characteristic of a field?

  1. The number of elements in the field

  2. The smallest positive integer such that $1 + 1 + ... + 1 = 0$

  3. The largest prime factor of the order of the field

  4. The number of generators of the field

Show answer
Correct Option: B
Explanation:

The characteristic of a field is the smallest positive integer such that $1 + 1 + ... + 1 = 0$, where the sum is taken over the field. In the set of real numbers, the characteristic is 0.

Which of the following is not a field?

  1. The set of rational numbers

  2. The set of real numbers

  3. The set of complex numbers

  4. The set of quaternions

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Correct Option: D
Explanation:

The set of quaternions is not a field because it does not satisfy the associative property of multiplication.

What is the order of a field?

  1. The number of elements in the field

  2. The smallest positive integer such that $1 + 1 + ... + 1 = 0$

  3. The largest prime factor of the order of the field

  4. The number of generators of the field

Show answer
Correct Option: A
Explanation:

The order of a field is the number of elements in the field.

What is a generator of a field?

  1. An element of the field that generates the entire field under addition

  2. An element of the field that generates the entire field under multiplication

  3. An element of the field that generates the entire field under both addition and multiplication

  4. An element of the field that generates the entire field under neither addition nor multiplication

Show answer
Correct Option: C
Explanation:

A generator of a field is an element of the field that generates the entire field under both addition and multiplication.

Which of the following is a finite field?

  1. The set of rational numbers

  2. The set of real numbers

  3. The set of complex numbers

  4. The set of integers modulo 7

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Correct Option: D
Explanation:

A finite field is a field with a finite number of elements. The set of integers modulo 7 is a finite field with 7 elements.

What is the Galois group of a field?

  1. The group of automorphisms of the field

  2. The group of units of the field

  3. The group of generators of the field

  4. The group of elements of the field that have a multiplicative inverse

Show answer
Correct Option: A
Explanation:

The Galois group of a field is the group of automorphisms of the field.

What is the fundamental theorem of Galois theory?

  1. Every finite field is a Galois field

  2. Every Galois field is a finite field

  3. The Galois group of a field is isomorphic to the group of permutations of the roots of a polynomial

  4. The Galois group of a field is isomorphic to the group of automorphisms of the field

Show answer
Correct Option: C
Explanation:

The fundamental theorem of Galois theory states that the Galois group of a field is isomorphic to the group of permutations of the roots of a polynomial.

Which of the following is a Galois field?

  1. The set of rational numbers

  2. The set of real numbers

  3. The set of complex numbers

  4. The set of integers modulo 7

Show answer
Correct Option: D
Explanation:

A Galois field is a finite field that is also a Galois extension of its prime subfield. The set of integers modulo 7 is a Galois field because it is a finite field and it is a Galois extension of the field of integers modulo 2.

What is a primitive element of a finite field?

  1. An element of the field that generates the entire field under addition

  2. An element of the field that generates the entire field under multiplication

  3. An element of the field that generates the entire field under both addition and multiplication

  4. An element of the field that generates the entire field under neither addition nor multiplication

Show answer
Correct Option: B
Explanation:

A primitive element of a finite field is an element of the field that generates the entire field under multiplication.

Which of the following is not a property of a field?

  1. The field contains a multiplicative identity

  2. The field contains an additive inverse for every element

  3. The field is closed under addition and multiplication

  4. The field is ordered

Show answer
Correct Option: D
Explanation:

A field is not required to be ordered. For example, the set of complex numbers is a field, but it is not ordered.

What is the Wedderburn-Artin theorem?

  1. Every finite division ring is a field

  2. Every finite field is a division ring

  3. Every division ring is a field

  4. Every field is a division ring

Show answer
Correct Option: A
Explanation:

The Wedderburn-Artin theorem states that every finite division ring is a field.

Which of the following is not a field axiom?

  1. The field contains a multiplicative identity

  2. The field contains an additive inverse for every element

  3. The field is closed under addition and multiplication

  4. The field is associative under addition

Show answer
Correct Option: D
Explanation:

The field axioms do not require the field to be associative under addition. However, all known fields are associative under addition.

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