Adv Alegbra HW Solutions 104

Adv Alegbra HW Solutions 104 - the (noncommutative) ring of...

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104 Conversely, if r and s are units, then there are a R and b S with ra = 1 and sb = 1. Therefore, ( r , s )( a , b ) = ( 1 , 1 ) , and so ( r , s ) is a unit in R × S . It follows easily that ( r , s ) U ( R × S ) if and only if ( r , s ) U ( R ) × U ( S ) . (ii) Show that if m and n are relatively prime, then I mn = I m × I n as rings. Solution. The isomorphism ϕ : I mn I m × I n of additive groups in Theorem 2.128, given by [ a ] 7→ ( [ a ] m , [ a ] n ) [where [ a ] m de- notes the congruence class of a mod m ],is easily seen to preserve multiplication: [ ab ] m =[ a ] m [ b ] m and [ ab ] n =[ a ] n [ b ] n . (iii) Use part (ii) to give a new proof of Corollary 3.54: if ( m , n ) = 1, then φ( mn ) = φ( m )φ( n ) , where φ is the Euler φ -function. Solution. If ( m , n ) = 1, then ( U ( I mn ) = U ( I m × I n ) , by part (i), and so U ( I m × I n ) = U ( I m ) × U ( I n ). But φ( m ) =| U ( I m ) | . and so φ( mn ) =| U ( I mn ) |=| U ( I m ) × U ( I n ) | =| U ( I m ) || U ( I n ) |= φ( m )φ( n ). 3.55 (i) Prove that the set F of all 2 × 2 real matrices of the form A = £ ab ba ± is a f eld with operations matrix addition and matrix mul- tiplication. Solution. It is easy to check that F is a commutative subring of the (noncommutative) ring of all 2
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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 is a homomor-phism 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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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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