the field remembers
the string is not empty
A set \(F\) with operations \(+\), \(\times\), satisfying closure, associativity, commutativity, identities, inverses,
and distributivity.
Given a finite nonempty set of symbols called an alphabet \( \Sigma \), a string (or word) over \( \Sigma \) is a finite ordered sequence of symbols chosen from \( \Sigma \).