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This appears in Dickson's "Linear Groups with an exposition of the Galois field theory", page 50, chapter 4.

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My question is: how would the above "translate" in modern terms? In particular, do we have a name for this multiplier Galois field?

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"Scalar" perhaps? – anon Apr 1 '12 at 9:09
I haven't included what Dickson writes next, but he goes on to show that these "multipliers" actually form a subfield of $F_{p^n}$ which he calls the multiplier Galois field of the additive group $[\lambda_1,...,\lambda_m]$. Now that I think of it, it's actually this thing I wanted to ask about, whether it has a special name that is. – the_fox Apr 1 '12 at 9:21
Just edited pictures and question. – the_fox Apr 1 '12 at 9:28
What does Dickson do with this concept later? His term "marks" is an old way of saying "elements". He first defines the additive group $V = {\mathbf F}_p^m$, and to make sense of multiplying elements of $V$ by elements of ${\mathbf F}_{p^n}$ in a natural way it seems like one should view $V$ inside $W = {\mathbf F}_{p^n}^m$. But then the only $\mu$ that fit his condition would be elements of ${\mathbf F}_p$, which can't be what he intends. Hmm. What he is doing seems similar to the definition of the multiplier of a lattice in algebraic number theory: if $K$ is a number field (contd..) – KCd Apr 1 '12 at 12:09
of degree $n$ over ${\mathbf Q}$ and $M$ is a full lattice in $K$ (meaning a ${\mathbf Z}$-module of rank $n$ in $K$) then the multiplier ring of $M$ is ${\mathcal O}_M = \{\alpha \in K : \alpha{M} \subset M\}$. This is an order in $K$ (not necessarily the maximal order = alg. integers of $K$) and it's a useful concept. Back in Dickson's setting, his condition ${\mu}V \subset V$ looks like $\alpha{M} \subset M$, so you can a similarity, but I already indicated that it seems like Dickson's thing is just leading to ${\mathbf F}_p$, which has to be a misunderstanding on my part. – KCd Apr 1 '12 at 12:13

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