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1. Normal matrix that isn't diagonalizable; counterexample?

I've been reading that the diagonalizable matrices are normal, that is, they commute with their adjoint: ##M^*M=MM^*##, where ##M^*## is the conjugate transpose of ##M##. So a matrix is diagonalizable if and only if it is normal, see: http://en.wikipedia.org/wiki/Normal_matrix But from...
2. Comparing definitions of groups, rings, modules, monoid rings

Hi, I wanted to see what people think about my current viewpoint on recognizing structures in abstract algebra. You count the number of sets, and the number of operations for each set. You can also think about action by scalar or basis vectors. So monoids groups and rings have one set...
3. Significance of difference between ring and algebra

(MAJOR EDIT: I think I missed the associative part, is that more or less my only mistake?) I've got an un"well-formed" question, I've been staring at things like every ring is a module over itself, counting the number of sets and operations in various definitions of algebraic objects. I was...
4. Is it called ring because of a clock?

Hi, does anyone know why they call a ring a ring. Was it because of Z/(n), where the numbers sort of form a ring in sense? I'm visualizing Z/(n) as a circle like 1 thru 12 on a clock. Or {0,1,2,...,11} if you prefer.
5. Why do morphisms in category of rings respect identity

Hi, I'm looking for intuition and/or logic as to why we would want or need morphisms according to axioms in category theory, to imply that in the category of rings, they must preserve the identity (unless codomain is "0"). Thank you very much.
6. Galois in a nutshell

Hi I struggled with Galois theory as a senior, and would like to improve my ability to jump in and out of the subject with a sort of big picture to details map. So as I see it, there are four things to be proficient at: Knowledge and understanding of: Theory, Particular examples, Relevant...
7. When is a Galois group not faithful

Hi, I'm looking at proposition 1.14(c) of Artin's Algebra. It says if we have K a splitting field for polynomial f from F[x], with roots a_1,...,a_n, then the Galois group G(K/F) acts faithfully on the set of roots. I look at faithful as the symmetries in the roots completely represent...