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A polynomial $p$ over a field $k$ is called irreducible if $p=fg$ for polynomials $f,g$ implies $f$ or $g$ are constant. One can consider the determinant of an $n\times n$ matrix to be a polynomial in $n^2$ variables. Does anyone know of a slick way to prove this polynomial is irreducible?

It feels like this should follow quite easily from basic properties of the determinant or an induction argument, but I cannot think of a nice proof. One consequence of this fact is that $GL_n$ is the complement of a hypersurface in $M_{n}$. Thanks.

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    $\begingroup$ 1) Can you represent a determinant in terms of smaller determinants? 2) Actually, $\mathrm{GL}_n$ is the complement to a hypersurface, although it can be made one in higher dimension. $\endgroup$ Commented May 19, 2012 at 15:52
  • $\begingroup$ Thanks for the hypersurface remark, it has been corrected. You can of course represent the determinant in terms of smaller determinants, but given something like $det_n = x_{11}det_{n-1,11} + ... + x_{1n}det_{n-1,1n}$ (with obvious notation used) it's not clear how to use this to show irreducibility as the $det_{n-1,1i}$ contain many of the same variables. $\endgroup$
    – user31559
    Commented May 19, 2012 at 15:58
  • $\begingroup$ Is slick a mathematical term I'm not aware of? $\endgroup$
    – corsiKa
    Commented May 19, 2012 at 20:04
  • $\begingroup$ I meant slick as in 'elegant' or 'nice'. Perhaps it's not a common phrase universally. For example, one can interpret the determinant as a change in volume, so perhaps one could prove the irreducibility using some geometric argument. $\endgroup$
    – user31559
    Commented May 20, 2012 at 0:25

4 Answers 4

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Deonote $p$ the determinant polyonomial. Observing that $p$ is of degree one in $x_{ij}$ for every $(i,j)$.

Now we can prove $p$ is irreducible. Suppose $p=fg$. Consider $x_{11}$. Suppose $x_{11}$ appears in $f$, then $f$ is of degree one in $x_{11}$ and $g$ is of degree zero in $x_{11}$. Now consider $x_{1j}$, then $x_{1j}$ must appear in $f$, otherwise $g$ is of degree one in $x_{1j}$ and $f$ is degree zero in $x_{1j}$, then the equality $$fg=(ax_{11}+b)(cx_{1j}+d)=acx_{11}x_{1j}+bcx_{1j}+adx_{11}+bd\in \mathbb{F}[x_{11}, x_{1j}, \dots]$$ leads to contradiction. So all $x_{1j}$ in $f$ for $j=1,\ldots,n$. Similar $x_{j1}$ are all in $f$. And since $x_{j1}$ is in $f$, it follows $x_{jk}$ are in $f$. Finally, all $x_{ij}$ are in $f$. And $g$ is a constant. We are done!


Edit: Contradiction: view $p$ be a polynomial of $x_{11},x_{1j}$, then $$p=x_{11}h_1+x_{1j}h_2+h_3\in \mathbb{F}[x_{11},x_{1j}, \dots],$$ where $h_1,h_2,h_3 \in \mathbb{F}[\{x_{ij}\}\mid x_{ij}\neq x_{11},x_{1j}]$, i.e., they are "constant" about $x_{11},x_{1j}$, but $$fg=acx_{11}x_{1j}+bcx_{1j}+adx_{11}+bd,$$ while $0\neq ac \in \mathbb{F}[\{x_{ij}\}\mid x_{ij}\neq x_{11},x_{1j}]$ and $bc,ad,bd$ are "constant" about $x_{11},x_{1j}$(all the results come from the assumption $f$ is a polynomial of degree one in $x_{11}$ and of degree zero in $x_{1j}$ and $g$ is of degree one in $x_{1j}$ and of degree zero in $x_{11}$), so $p$ cannot equal to $fg$ since the definition of the determinant.

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  • $\begingroup$ Dear wxu, could you explain why there is a contradiction ? $\endgroup$
    – user18119
    Commented May 19, 2012 at 19:39
  • $\begingroup$ Thanks! I'll accept your answer in a few days if no one else has a more elegant proof (your proof is quite nice). $\endgroup$
    – user31559
    Commented May 20, 2012 at 0:26
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    $\begingroup$ @wxu: I see. You mean $p$ has total degree one when viewed as polynomial in $x_{11}, x_{1j}$. Nice proof ! $\endgroup$
    – user18119
    Commented May 20, 2012 at 6:35
  • $\begingroup$ this is a nice proof which also help me this the problem that I posted today, thanks a lot.@wxu $\endgroup$
    – user53800
    Commented Dec 27, 2012 at 21:46
  • $\begingroup$ Why can't $g$ contain $x_{11}$? Perhaps the product of the terms in $f$ and $g$ containing $x_{11}$ gets cancelled out by another such product. $\endgroup$
    – Tri
    Commented Oct 14, 2021 at 6:57
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This answer is basically a proof from M.Bocher "Introduction to higher algebra" (Dover 2004) on pages 176-7.

A better link is here

Let me sketch what's there. Argue by contradiction. Let the determinant $D=\det(a_{ij})_{i,j}$ be the product $D=fg$ of non-constant in $a_{ij}$ polynomials $f$, $g$.

Note that $D$ involves $a_{ij}$ for any $1\leq i,j\leq n$, and it is linear in $a_{ij}$, as can be seen by expanding $D$ w.r.t. the $i$-th row. W.l.o.g. $f$ is linear in $a_{ii}$, and $g$ does not depend on $a_{ii}$. Now, $g$ cannot depend on $a_{ij}$ for any $j\neq i$, as this would imply that $D$ contains a term divisible by $a_{ii}a_{ij}$ - impossible by definition of the determinant. Thus $g$ does not depend on any $a_{ij}$, and, by symmetry, $a_{ji}$, $1\leq j\leq n$. Thus $f$ depends linearly on all the $a_{ij}$, and $a_{ji}$, $1\leq j\leq n$.

Therefore if $g$ is not constant it must depend linearly on $a_{jj}$, for some $j\neq i$. But then $a_{ij}$ must occur in both $f$ and $g$, the final contradiction.

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Here is a proof using a little bit of algebraic geometry (we assume k is algebrically closed). It can be seen that $V(det)$ is irreducible in $k^{n^2}$. It is for example the image of the morphism $M_n(k)\times M_n(k) \rightarrow V(det)$, $(P,Q)\mapsto PI_{n-1}Q$ and the source is irreducible.

Hence as $k[X_{ij}]$ is a UFD we have $I(V(det)) = \sqrt{(p)}$ where p is an irreducible polynomial. So $det = p^k$.

To show that $k=1$ we use the fact that the differential of the determinant at a matrice A is $D(det)_A(.) = Tr(Adj(A).)$ where Adj(A) is the adjugate matrix. For $A$ such that $rk(A)=n-1$ we have $rk(Adj(A))=1$ so $D(det)_A(.)\neq 0$ which forces $k=1$.

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  • $\begingroup$ How can you multiply $P$ and $I_{n-1}$, since these matrices have different sizes? $\endgroup$ Commented Jan 27, 2022 at 17:45
  • $\begingroup$ @GeorgesElencwajg: I guess Moustik means $I_{n-1} \oplus 0 \cdot I_1$ $\endgroup$
    – math54321
    Commented Aug 14, 2022 at 20:43
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Here is another algebraic argument which may be of interest, using induction on the size of the matrix:

Let $A_{ij}$ be the minor we get when we delete row $i$ and column $j$. If we expand along the first row of the matrix, we get $$\det A=(\det A_{11})x_{11}-(\det A_{12})x_{12}+\cdots\pm(\det A_{1n})x_{1n}.$$ None of the determinants $\det A_{1i}$ contains any variable $x_{1i}$, so this is a linear polynomial in the variables $x_{1i}$. Therefore it can only factor if the coefficients of the $x_{1i}$ have a common factor. But these coefficients are determinants of size $n-1$ so by induction they are all irreducible; since they are not scalar multiples of each other they have no (non-constant) common factor.

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