I need help with abstract algebra

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Senapboebf

Mathematics

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Heres what GL stands for




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0 1 0 Consider the following element 0 0 1 of GL(3,22). 1 1 0 Compute all of its powers. How many elements would you have to add for this set of powers to be closed under addition? Find the characteristic polynomial of each of the powers. Find the lowest degree polynomial that all of the powers satisfy. Have you constructed a field? BONUS: Show that every irreducible cubic over Z, has a root among these powers. Example 3. The Group of Units of M,(Q), and Matrix Rings in General. The invertible elements in M,(Q) are the rational matrices with nonzero determinant (as is well-known) or in other words, the General Linear Group GL(n,Q). Similarly for M,(R), the group is GL(n,R) and for M, (C), it is Gl(n,C).
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Very useful material for studying!

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