(i) Let A be a four-dimensional vector space over ℚ , and let 1, i, j, k be a basis.

step 1 Okay, so we're looking at question 12 .4, and we're asked to prove the theorem of theorem 12 .1.

(i) Let A be a four-dimensional vector space over ℚ ,...

{ (i) Let } A \text { be a four-dimensional vector space over } \mathbb{Q} \text {, and let } \\ & 1, i, j, k \text { be a basis. Prove that } A \text { is a division algebra over }\end{array}$ $\mathbb{Q}$ if we define 1 to be the identity and $$ \begin{array}{lll} i^{2}=-1, & j^{2}=-2, & k^{2}=-2, \\ i j=k, & j k=2 i, & k i=j, \\ j i=-k, & k j=-2 i, & i k=-j . \end{array} $$ (ii) Prove that $\mathbb{Q}(i)$ and $\mathbb{Q}(j)$ are nonisomorphic maximal subfields of $A$.

{ (i) Let } A \text { be a four-dimensional vector space over } \mathbb{Q} \text {, and let } \\ & 1, i, j, k \text { be a basis. Prove that } A \text { is a division algebra over }\end{array}$ $\mathbb{Q}$ if we define 1 to be the identity and
$$
\begin{array}{lll}
i^{2}=-1, & j^{2}=-2, & k^{2}=-2, \\
i j=k, & j k=2 i, & k i=j, \\
j i=-k, & k j=-2 i, & i k=-j .
\end{array}
$$
(ii) Prove that $\mathbb{Q}(i)$ and $\mathbb{Q}(j)$ are nonisomorphic maximal subfields of $A$.

Key Concepts

Division Algebra

A division algebra is an algebraic structure in which every nonzero element has a multiplicative inverse, ensuring that there are no zero divisors. This property is vital because it generalizes the idea of a field to situations where multiplication may not be commutative. Proving that an algebra is a division algebra typically involves showing that the product of any two nonzero elements is nonzero and that an inverse exists for each nonzero element.

Quaternion-like Algebras

Quaternion-like algebras are examples of noncommutative algebras that generalize the classical quaternions via a prescribed set of multiplication rules on a basis. They are typically defined over a field, and their multiplication is structured so that certain basis elements square to elements of the field (often negative numbers) and interact according to specific relations. Verifying these algebras are division algebras involves careful computation to ensure the absence of zero divisors, even when the algebra is noncommutative.

Maximal Subfields

A maximal subfield of an algebra is a subfield that is not properly contained in any larger subfield of the algebra. In the context of a division algebra, maximal subfields are significant because they provide deep insight into the algebra's structure and often have the highest degree possible over the center of the algebra. Demonstrating maximality typically means showing that any subfield containing it would necessarily coincide with the entire algebra.

Field Extensions

Field extensions arise when a smaller field is augmented by adding new elements, resulting in a larger field. In division algebras, many important substructures appear as field extensions of the center of the algebra. Studying these field extensions can reveal information about the structure and properties of the division algebra, such as dimensions and the interplay between the subfields and the entire algebra.

Isomorphism of Fields

Field isomorphism refers to a bijective mapping between fields that preserves the operations of addition and multiplication. Establishing whether two subfields of an algebra are isomorphic involves comparing their algebraic structures—such as the behavior of their elements under multiplication—and determining whether they can be mapped onto one another while preserving all field operations. Distinguishing nonisomorphic maximal subfields highlights the diverse field structures that can coexist within a single algebraic system.

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