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Using these monomial bases we prove that the image of the transfer for a general linear group over a finite field is a principal ideal in the ring of invariants.
      
Using related sequences of Lucas numbers, other 3-manifolds are constructed, their geometric structures determined, and a curious relationship between the homology and the invariant trace-field examined.
      
We consider varieties over an algebraically closed field k of characteristicp>amp;gt;0.
      
LetG be a classical algebraic group defined over an algebraically closed field.
      
The kernel of a certain derivation of the polynomial ringk[6] is shown to be nonfinitely generated overk (a field of charactersitic zero), thus giving a new counterexample to Hilbert's Fourteenth Problem.
      
For example, if our groupG isSn, these objects are field extensions; ifG=On, they are quadratic forms; ifG=PGLn, they are division algebras (all of degreen); ifG=G2, they are octonion algebras; ifG=F4, they are exceptional Jordan algebras.
      
We prove that these determinantal semi-invariants span the space of all semi-invariants for any quiver and any infinite base field.
      
This is true regardless of the characteristic of the field or of the order of the parameterq in the definition ofHn.
      
Let ρ:G?Gl(n,) be a representation of a finite groupG over a field such that the ring of invariants is a polynomial algebra.
      
We show that the absolute invariants (i.e.,the ${\mbox{\rm GL}}(2, {\mbox{\bf R}})$-invariants in the field of fractions of ${\mathcal P}$) distinguish the isomorphism classes of 2-dimensional non-associative real division algebras.
      
Let $X$ be a smooth projective curve over the field of complex numbers, and fix a homogeneous representation $\rho\colon \mathop{\rm GL}(r)\rightarrow \mathop{\rm GL}(V)$.
      
Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0.
      
In case p >amp;gt; 0, assume G is defined and split over the finite field of p elements Fp.
      
Let k be a perfect field and G an algebraic group defined over k.
      
Finite complex reflection groups have the remarkable property that the character field k of their reflection representation is a splitting field, that is, every irreducible complex representation can be realized over k.
      
Let k be a field of characteristic zero, let a,b,c be relatively prime positive integers, and define a
      
Let G be a connected and reductive group over the algebraically closed field K.
      
Let k be an algebraically closed field of characteristic p ≥ 0.
      
For the special case of his construction that gives groups of type E6, we connect the two papers by answering the question: Given an Albert algebra A and a separable quadratic field extension K, what is the index of the resulting algebraic group?
      
Let p be a prime and let V be a finite-dimensional vector space over the field $\mathbb{F}_p$.
      
 

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