Let D be the ℚ-subalgebra of ℍ having basis 1, i, j, k. (i) Prove that D is a division algebra o

Let D be the ℚ-subalgebra of ℍ having basis 1, i, j, k. (i)...

Key Concepts

Division Algebra

A division algebra is an algebra in which every nonzero element has a multiplicative inverse. This concept generalizes the idea of a field by allowing noncommutative multiplication while still requiring that the division property holds. The study of division algebras is central in algebra because it provides insights into the structure and classification of algebras over a given field, such as the real numbers or the rational numbers.

Center of an Algebra

The center of an algebra is the subset consisting of elements that commute with every other element in the algebra. Understanding the center is crucial because it often acts as the underlying scalar field over which the algebra is defined. In many cases, for instance in quaternion algebras, computing the center can show that the algebra is central simple, which has important implications for its structure and the existence of certain subfields.

Quaternion Algebra

A quaternion algebra is a four-dimensional algebra over a field which is generated by elements that satisfy specific multiplicative relations. It is an example of a noncommutative division algebra when every nonzero element is invertible. Quaternion algebras are significant in algebra and number theory as they provide concrete examples of division algebras and have deep connections with quadratic forms and the theory of central simple algebras.

Maximal Subfield

A maximal subfield of a division algebra is a subfield that is not properly contained in any larger subfield of the algebra. In the context of a quaternion algebra, maximal subfields are often quadratic field extensions over the center, and they capture essential information about the algebra’s splitting fields and structure. Studying maximal subfields helps in understanding the embedding of simpler algebraic structures within more complex ones.

Quadratic Field Extension

A quadratic field extension is a field extension of degree two over a given field, typically constructed by adjoining a square root of an element not already a square in the base field. This concept is fundamental in algebra and number theory as it provides the simplest nontrivial examples of field extensions. In division algebras, such as quaternion algebras, quadratic field extensions serve as natural candidates for maximal subfields and often help in decomposing or analyzing the algebra's structure.

Transcript

00:01 Okay, so suppose we're given three polynomials, x squared plus 1, x squared minus 1, and 2x minus 1.

00:13 Okay, and i want to show that these three polynomials form a basis for p2.

00:24 Okay, so how would we go about showing something like this? well, the first step that we would do is we'll transform, okay, these polynomials into a form that is more familiar to us.

00:37 We'll write them directly as column vectors, so transform to vectors.

00:43 And then once we have done that step, then to show that it is a basis for p2 is the same thing as showing that a set of vectors is a basis for, let's say, r3.

00:55 And then show vectors, form basis.

01:02 So that's the step that we'll take.

01:04 So how do we transform? these polynomials into a more familiar vector form.

01:11 So consider the following row vector 1 x squared.

01:17 Ok, so we can ask, ok, what is the column vector that we have to multiply this row vector by in order to get x squared plus 1? well, that column vector is going to be 1 ,0 .1.

01:33 The second column vector, ok, x squared minus 1.

01:36 That's going to be minus one, zero, one, right? and then the third column vector that we'll need in order to get a 2x minus one, that's going to be equal to 0 to minus one, oh sorry, typo.

01:56 So the top one is going to be minus one, and then two, and then zero...

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