Unformatted text preview: the (noncommutative) ring of all 2 × 2 real matrices (note that the identity matrix I ∈ F ). If A ±= 0, then det ( A ) = a 2 + b 2 ±= 0, and so A − 1 exists; since A − 1 has the correct form, it lies in F , and so F is a f eld. (ii) Prove that F is isomorphic to C . Solution. It is straightforward to check that ϕ is a homomorphism of f elds; it is an isomorphism because its inverse is given by a + ib 7→ A . 3.56 True or false with reasons. (i) If a ( x ), b ( x ) ∈ F 5 [ x ] with b ( x ) ±= 0, then there exist c ( x ), d ( x ) ∈ F 5 [ x ] with a ( x ) = b ( x ) c ( x ) + d ( x ) , where either d ( x ) = 0 or deg ( d ) < deg ( b ) . Solution. True....
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 Fall '11
 KeithCornell
 Vector Space, Ring

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