Question

I don’t know how to do this question.

(4) (a) Define a skew-field by the following analogy: commutative ring field ring skew-field What distinguishes fields from skew-fields? (b) Find an example of a skew-field on the internet and learn how to compute the multiplication in this skew-field. Hint: The quaternions are a good example to read about.

Answer 1

The dollowing analogy commutative ring: field: : ring: stewfiness this show A commutative ring R is said to be a fiels if they non zero element of R has multiplicative increases and if we say ring is not commutation only ring , then we define akew fiels implies A non-commutation ring R is said to be skew fiels\d is every non zero element of R has multiplication inverse i.e, field implies commuatative ring skewfield implies ring only not ne\ecssarly commutative note: Every field is skew field but skewfiels nud not be a field b) Let R = {a + b i + c j + d k: a, b, c, d elementofteta and i^2 = j^2 = k^2 = 1 I j = k, j I = -k} (R + .) is a skew fiels but not field because hence multiplication is not commutative I . j = k j . I = -k