What Do You Know About Hurwitz's Theorem On Composition Algebras?

1) When was the theorem published posthumously?

1922

1923

1924

1925

The theorem was published posthumously in 1923.

Explanation

The theorem was published posthumously in 1923.

2) Which of these is the correct representation of the theorem?

Q ( a b) = q ( a) q ( b)

Q ( a b) = q ( a) / q ( b)

Q ( a b) = q ( a) ( b)

Q ( a b) = ( a) ( b)

The correct representation of the theorem is q ( a b) = q ( a) q ( b) because it follows the pattern of multiplying the values of q ( a) and q ( b) together. The other options do not follow this pattern and do not correctly represent the theorem.

Explanation

The correct representation of the theorem is q ( a b) = q ( a) q ( b) because it follows the pattern of multiplying the values of q ( a) and q ( b) together. The other options do not follow this pattern and do not correctly represent the theorem.

3) The Hurwitz algebras are an example of what?

Composition algebras

Non-composition algebras

Real algebras

Irrational algebras

The Hurwitz algebras are a specific type of composition algebras. Composition algebras are a type of algebraic structure that generalize the properties of the real numbers, complex numbers, and quaternions. The Hurwitz algebras satisfy the properties of composition algebras, making them an example of this type of algebraic structure.

Explanation

The Hurwitz algebras are a specific type of composition algebras. Composition algebras are a type of algebraic structure that generalize the properties of the real numbers, complex numbers, and quaternions. The Hurwitz algebras satisfy the properties of composition algebras, making them an example of this type of algebraic structure.

4) Which form defines the hormomorphism?

Quadratic form

Algebraic form

Exponential form

Geometric form

The quadratic form is the form that defines the hormomorphism. This form is a polynomial function of degree two, where the variables are multiplied by themselves. It is commonly used in algebra and has various applications in fields such as physics and optimization. The other forms mentioned, algebraic form, exponential form, and geometric form, do not specifically define the hormomorphism.

Explanation

The quadratic form is the form that defines the hormomorphism. This form is a polynomial function of degree two, where the variables are multiplied by themselves. It is commonly used in algebra and has various applications in fields such as physics and optimization. The other forms mentioned, algebraic form, exponential form, and geometric form, do not specifically define the hormomorphism.

5) The multiplicative formulas used for the sum of squares can occur in each of the following dimensions except which one?

1

2

4

5

The multiplicative formulas used for the sum of squares can occur in dimensions 1, 2, and 4. However, it cannot occur in dimension 5.

Explanation

The multiplicative formulas used for the sum of squares can occur in dimensions 1, 2, and 4. However, it cannot occur in dimension 5.

6) Which of these has the theorem never been applied to?

Classical group

Crystallized group

Algebraic topography

Homotopy group

The theorem has been applied to classical groups, algebraic topography, and homotopy groups. However, there is no record or evidence of the theorem ever being applied to a crystallized group.

Explanation

The theorem has been applied to classical groups, algebraic topography, and homotopy groups. However, there is no record or evidence of the theorem ever being applied to a crystallized group.

7) The hormomorphism is defined into what?

Positive real number

Negative real number

Integer

Complex number

The question is asking about the definition of hormomorphism. A hormomorphism is defined as a function that preserves the order and the operations of addition and multiplication. Since the options given are positive real number, negative real number, integer, and complex number, the only option that fits the definition of hormomorphism is positive real number.

Explanation

The question is asking about the definition of hormomorphism. A hormomorphism is defined as a function that preserves the order and the operations of addition and multiplication. Since the options given are positive real number, negative real number, integer, and complex number, the only option that fits the definition of hormomorphism is positive real number.

8) Which part of the algebra is involved in the theorem?

Zero part

Irrational numbers

Non-zero part

Rational numbers

The theorem involves the non-zero part of algebra. This suggests that the theorem is applicable to elements of algebra that are not equal to zero. It implies that the theorem does not apply to the zero element or any operations involving it.

Explanation

The theorem involves the non-zero part of algebra. This suggests that the theorem is applicable to elements of algebra that are not equal to zero. It implies that the theorem does not apply to the zero element or any operations involving it.

9) According to the theorem, the algebra is NOT isomorphic to which of the following?

Real numbers

Complex numbers

Integers

Octonions

The theorem states that the algebra is not isomorphic to the integers. Isomorphism is a concept in algebra that describes a one-to-one correspondence between two algebraic structures, preserving their operations and relations. In this case, the theorem suggests that the algebra being discussed is isomorphic to the real numbers, complex numbers, and octonions, but not to the integers.

Explanation

The theorem states that the algebra is not isomorphic to the integers. Isomorphism is a concept in algebra that describes a one-to-one correspondence between two algebraic structures, preserving their operations and relations. In this case, the theorem suggests that the algebra being discussed is isomorphic to the real numbers, complex numbers, and octonions, but not to the integers.

10) In the theorem, what does q represent?

Irrational numbers

Real positive numbers

Negative-definate

Positive-definate

The variable q in the theorem represents positive-definite numbers. This means that q represents numbers that are greater than zero and have a positive value. Positive-definite numbers are important in various mathematical fields, such as linear algebra and optimization, where they have specific properties and applications.

Explanation

The variable q in the theorem represents positive-definite numbers. This means that q represents numbers that are greater than zero and have a positive value. Positive-definite numbers are important in various mathematical fields, such as linear algebra and optimization, where they have specific properties and applications.