A non-linear automorphism of Let be a field. For any we let So the realtions that define become for all
Lemma 1. Let and Then
Proof. An easy induction shows that for all Applying this, we will get that if then So, since every element of is a finite linear combination of monomials in the form we will get
for all Both parts of the lemma are straightforwad results of
Notation. Let and fix an integer For every choose and put
Lemma 2. For any we have
Proof. If then and we are done. If then will not occur in and so
Now define the maps and on the generators by
for all and extend the definition homomorphically to the entire to get -algebra homomorphisms of Of course, we need to show that these maps are well-defined i.e. the images of under and satisfy the same relations that do. Before that, we prove an easy lemma.
Lemma 3. for all
Proof. Let where and Then
But by our choice and thus A similar argument shows that
Lemma 4. The maps and are well-defined.
Proof. I will only prove the lemma for because the proof for is identical. Since we have for all and thus and commute. The relations follow from the first part of Lemma 1 and Lemma 2. The relations follow from the second part of Lemma 1 and Lemma 2.
Theorem. The -algebra homomorphisms and are automorphisms.
Proof. We only need to show that and are the inverse of each other. Lemma 3 gives us and Lemma 2 with Lemma 3 will give us for all