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Results tagged with ra.rings-and-algebras
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user 4354
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
15
votes
The sum of same powers of all matrices modulo p
The sum is zero for all $k<p^2-1$.
Assume that $k$ is a multiple of $p-1$ and $k<p^2-1$. Divide all matrices into classes of the form $\{A,A+1,A+2,\dots,A+(p-1)\}$ . Summimg the $k$th powers over su …
- 32.4k
10
votes
Is there a field which is the union of finitely many proper subfields?
There are 3 cases.
Case 1. The field is finite. Then, as Charles Matthews pointed out, the primitive element theorem does the job.
Case 2: The intersection of the subfields is infinite. This is cove …
- 32.4k
7
votes
Accepted
Existence of definite symmetric matrices satisfying affine linear constraints
No. Consider the 3-dimensional affine subspace consisting of all matrices having $1,1,-1$ on the diagonal (non-diagonal entries are arbitrary). A small perturbation of this subspace cannot intersect t …
- 32.4k