A Galois theory example by Brian Osserman

By Brian Osserman

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Let ^ be a division ring. Prove that an element of the total m by m matrix ring 2m is nonsingular if and only if it is not a left (or right) divisor of zero. 2. Let<3be a division ring. Prove that an element of the total m by m matrix ring Q)m is nonsingular if and only if it is a product of matrices which are obtained from the unity matrix Im via transformations of type (i), (ii) or (iii) mentioned on page 33. 37 Linear Associative A Igebras 3. Find the invariant factors and the elementary divisors (over the ring of integers) of a 3 by 3 matrix {ati) such that atj = 1 for i,j = 1,2,3.

Clearly, (p is a homomorphism from ^ onto v? W - Therefore, as mentioned on page 46, W =

Is the field of real numbers, is such that its invariant factors are: U2+l)2(x-5)3, (x2 + l)(jt-5) 2 , 1, 1,0 then its elementary divisors are: (jt 2 +l) 2 , (x2+l), (x-5)\ (JC-5)3, 1, 1, 0. Conversely, if a matrix over 3F\x\ where 3P is the field of real numbers, is such that its elementary divisors are: (;t2 + l) 5 , (JC2 + 2) 3 , (x2 + 2), (Jt-1) 3 , 1, 0, 0 then its invariant factors are: (JC 2 +1) 5 (JC 2 + 2 ) 3 ( J C - 1 ) 3 , (JC2 + 2 ) , 1, 0, 0. Following Corollary 4 we have COROLLARY 6.

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