Prove that the product of homogeneous polynomials is homogeneous.
Let be a ring and let be homogeneous of degree and respectively; say and with and for all and , respectively.
Then . Note that for each , we have . Thus is homogeneous.
Prove that the product of homogeneous polynomials is homogeneous.
Let be a ring and let be homogeneous of degree and respectively; say and with and for all and , respectively.
Then . Note that for each , we have . Thus is homogeneous.
On these pages you will find a slowly growing (and poorly organized) list of proofs and examples in abstract algebra.
No doubt these pages are riddled with typos and errors in logic, and in many cases alternate strategies abound. When you find an error, or if anything is unclear, let me know and I will fix it.