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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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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. – Alexei Averchenko May 19 '12 at 15:52
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. – user31559 May 19 '12 at 15:58
Is slick a mathematical term I'm not aware of? – corsiKa May 19 '12 at 20:04
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. – user31559 May 20 '12 at 0:25
up vote 26 down vote accepted

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 $fg=(ax_{11}+b)(cx_{1j}+d)=acx_{11}x_{1j}+bcx_{1j}+adx_{11}+bd\in k[\ldots][x_{11},x_{1j}]$, 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 $p=x_{11}h_1+x_{1j}h_2+h_3\in k[\ldots][x_{11},x_{1j}]$, where $h_1,h_2,h_3 \in k[\{x_{ij}\}\mid x_{ij}\neq x_{11},x_{1j}]$, i.e., they are "constant" about $x_{11},x_{1j}$, but $fg=acx_{11}x_{1j}+bcx_{1j}+adx_{11}+bd$, while $0\neq ac \in k[\{x_{ij}\}\mid x_{ij}\neq x_{11},x_{1j}]$ and $bc,ad,bd$ are "constant" about $x_{11},x_{1j}$(all the results come from the assumption $f$ is a polynomila 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 $fg$.

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Dear wxu, could you explain why there is a contradiction ? – user18119 May 19 '12 at 19:39
Thanks! I'll accept your answer in a few days if no one else has a more elegant proof (your proof is quite nice). – user31559 May 20 '12 at 0:26
@wxu: I see. You mean $p$ has total degree one when viewed as polynomial in $x_{11}, x_{1j}$. Nice proof ! – user18119 May 20 '12 at 6:35
this is a nice proof which also help me this the problem that I posted today, thanks a lot.@wxu – user53800 Dec 27 '12 at 21:46

The proof above can be found in M.Bocher "Introduction to higher algebra" (Dover 2004) on pages 176-7.

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