Dummit and Foote Exercise 2, Section 13.2, page 529 reads as follows:

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2. Let and let . Obtain fields of 4, 8, 9 and 27 elements by adjoining a root of f(x) to a field F where f(x) = g(x) or h(x) and or . Write down the multiplication tables for for the fields with 4 and 9 elements and show that the non-zero elements form a cyclic group.

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In my first attempt at this exercise I took and

So we have and f(x) as above.

The elements of , then, are as follows:(see attachment)

,

,

,

and

The multiplication table can then be composed using the following:

That is ... ... ... ... ... (1)

So the multiplication table, composed using (1) is as follows:(see attachment)

So far so good (i think?) but when I test whether the 9 non-zero elements in the multiplication table form a cyclic group they do not

For example if you try a cyclic group for the non-zero elements based on we find:

(?? should generate another member of the group but does not!)

and

(?? not an element of the group)

Can anyone help?

Peter