1) If is a morphism then see:
http://www.artofproblemsolving.com/Foru ... 0&t=218429 I have not reread my first post but I think that it shows that where is a morphism of .
2) Now let be a morphism.
Case 1: then and we are done.
Case 2: . If is invertible then necessarily .
Let be a non invertible matrix.
Case 2.1: . If is nilpotent then .
is product of nilpotent matrices (Sullivan,Product of nilpotent matrices,Linear and multilinear algebra,vol 56,3/2008/ p. 311-317). This result is valid for and I think, for any field.
Therefore . We are done.
Case 2.2: . There exists s.t. ;then . Thus if is idempotent then . is a product of idempotent (for any field): see
http://www.artofproblemsolving.com/Foru ... 0&t=271857 thus .
Let be an invertible matrix. . Therefore and we are done.
Conclusion: factors through the determinant without supplementary hypothesis.
PS: it remains to solve the case 2.1 with . I go to bed.
_________________ Le vieux loup perd ses poils mais garde son vice. Hard work beats talent when talent fails to work hard.
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