Algebra for Digital Communication
... (3) Let’s give explicit descriptions of these two homomorphisms, constructed, as usual, by sending [1] (in Z/4Z or Z/12Z) on 1R = [9]12 . Then using additivity, the only possibility is: f ([x]4 ) = f (x · [1]4 ) = x · f ([1]4 ) = x · [9]12 = [9x]12 , and g([x]12 ) = [9x]12 . We can then verify that ...
... (3) Let’s give explicit descriptions of these two homomorphisms, constructed, as usual, by sending [1] (in Z/4Z or Z/12Z) on 1R = [9]12 . Then using additivity, the only possibility is: f ([x]4 ) = f (x · [1]4 ) = x · f ([1]4 ) = x · [9]12 = [9x]12 , and g([x]12 ) = [9x]12 . We can then verify that ...
Basic reference for the course - D-MATH
... for all x, y ∈ F ∗ , x · y ∈ F ∗ . Also, 1 ∈ F ∗ . The field axioms imply (F ∗ , ·, 1) is an abelian group. The field axioms (i)-(iv) above may be rewritten in a simpler form: (i) (F, +, 0) is an abelian group, (ii) (F ∗ , ·, 1) is an abelian group, (iii) the distributive laws hold. The standard num ...
... for all x, y ∈ F ∗ , x · y ∈ F ∗ . Also, 1 ∈ F ∗ . The field axioms imply (F ∗ , ·, 1) is an abelian group. The field axioms (i)-(iv) above may be rewritten in a simpler form: (i) (F, +, 0) is an abelian group, (ii) (F ∗ , ·, 1) is an abelian group, (iii) the distributive laws hold. The standard num ...