Show element is invertible in a Clifford algebra

Here is the problem:

Let be a field and a quadratic space with an anisotropic form not representing (i.e. for all ). Show that if and are not both zero, then is an invertible element in the Clifford algebra .

I'm not really sure how to start this problem. I don't fully understand Clifford algebras, so I don't even really know what invertible elements even look like. I understand the definition - , where I is the ideal generated by - but I don't understand the resulting algebra (except when is a binary or nonzero unary form).

Any help is appreciated.

Re: Show element is invertible in a Clifford algebra

Quote:

Originally Posted by

**redsoxfan325** Here is the problem:

Let

be a field and

a quadratic space with an anisotropic form

not representing

. Show that if

and

are not both zero, then

is an invertible element in the Clifford algebra

.

I'm not really sure how to start this problem. I don't fully understand Clifford algebras, so I don't even really know what invertible elements even look like. I understand the definition -

, where I is the ideal generated by

- but I don't understand the resulting algebra (except when

is a binary or nonzero unary form).

Any help is appreciated.

what do u mean by "not representing 1"? do u mean for all ?

Re: Show element is invertible in a Clifford algebra

Quote:

Originally Posted by

**NonCommAlg** what do u mean by "not representing 1"? do u mean

for all

?

Yes.

Re: Show element is invertible in a Clifford algebra