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Wikipedia's article on the Leibniz determinant formula states there exists one and only function $F:M_n(\mathbb{K})\mapsto\mathbb{K}$ which is alternating, multilinear and $F(1)=1$. I assume $M_n$ is the set of $n\times n$ square matrices, build around some set or field... $\mathbb{K}$ - but what is $\mathbb{K}$?

Additionally, what is the difference between $\mapsto$ ("mapsto") and $\to$ ("to")?

TonyK
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FShrike
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    $\mathbb K$ can denote a generic field – J. W. Tanner Jun 22 '21 at 12:26
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    When you denote the domain and codomain of a function, you use $\to$ between them. When you denote an element in the domain, and its image in the codomain, you use $\mapsto$ between them. For instance, the function $\Bbb R\to\Bbb R$ given by $x\mapsto x^2$. – Arthur Jun 22 '21 at 12:26
  • Generally for things involving the determinant, we also require the characteristic of the field to not be 2, because otherwise alternating is meaningless – Alan Jun 22 '21 at 12:28
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    $K$ or $\Bbb K$ refers to the German word Körper, which means field. In English, often denoted by $F$. However, here the map is already denoted by $F$. – Dietrich Burde Jun 22 '21 at 12:28
  • @Alan I'm not well versed in fields - is this because a field characteristic two only ever has elements of 1 and 0, so the determinant goes to zero always anyway? – FShrike Jun 22 '21 at 12:30
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    No, the determinant then goes to $\Bbb F_2$, so $1$ or $0$. But alternating isn't saying much if $2=0$, so $1=-1$. – Dietrich Burde Jun 22 '21 at 12:31
  • Field : Deutsch Körper – Mauro ALLEGRANZA Jun 22 '21 at 12:32
  • @DietrichBurde thank you – FShrike Jun 22 '21 at 12:32
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    @Alan What?! No, determinant in characteristic $2$ is fine; you just have to state the alternating property correctly: a multilinear function $V^n\to W$ is alternating if $f(v_1,\cdots, v_n)=0$ whenever there are $i\ne j$ such that $v_i=v_j$. –  Jun 22 '21 at 12:47
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    @IdioticShrike Also there are infinitely many fields with characteristic 2, not only $\Bbb F_2$. – Janik Jun 22 '21 at 13:37

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$\mathbb K$ can denote a generic field (Körper in German),

though it would have been good if the Wikipedia article had defined it.

When there is a function, $\to$ indicates what domain is mapped to what codomain,

whereas $\mapsto$ indicates where it takes a particular element.

For example, $\det:M_n(\mathbb K)\to \mathbb K$, and $\det:I\mapsto1$.

J. W. Tanner
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