Irreducibility of Polynomials in $k[x,y]$

Assume $k$ is algebraically closed. Let $F = k((x))$. Consider the polynomial $g = xy - y^6$. Because

$$g \equiv f \mod x^6 $$

then over $F$, factorization of $g$ should approximate the factorization of $f$. The factorization of $g$ is

$$ g = y (x - y^5) $$

This is easy to check because its splitting field $E = F(x^{1/5})$ satisfies $[E : F] = 5$. The six roots of $g$ are $y=0$ and $y=\zeta^k x^{1/5}$ where $\zeta$ is a fifth root of unity. (if $k$ has characteristic $5$, then instead $x^{1/5}$ is a root with multiplicity 5)

Now observe

$$ f(x, x^{1/5}) \equiv g(x, x^{1/5}) = 0 \mod (x^{1/5})^{30} $$

and by Hensel's Lemma / Newton's algorithm, we can lift $x^{1/5}$ to an actual root $\alpha$ of $f$ in $E$ with leading term $x^{1/5}$; therefore $F(\alpha) = E$. (This might not work in characteristic 5; I haven't checked in detail)

Therefore, we conclude that $f$ has an irreducible factor of degree $5$ over $F$.

Therefore if $f$ were reducible over $k(x)$, then the only possibility is that it has an irreducible degree $5$ factor and a root. By inspection, we can see that such a root has to be $x^5$ plus higher powers of $x$, and from there it's easy to see that said root can't exist.

Thus $f$, as a polynomial in $y$ over $k[x]$, is irreducible and has content $1$. Therefore $f$ is irreducible in $k[x,y]$.

Now consider $k$ not algebraically closed, then the fact $f$ is irreducible over $\bar{k}[x,y]$ implies it is also irreducible over $k[x,y]$.

As a complement to other answers, I present here a very general method to see if a polynomial is irreducible of not. It needs a little more computation but it is not specific to the polynomial we consider.

There is an algorithm for absolute (i.e. over an algebraically closed field) factorisation of bivariate polynomials. It has been discovered by Picard (1906) and has been recently used by Ruppert (1999) and Gao (2003) to design efficient algorithms.

Let $f$ in $k[x,y]$ of bidegree $(m,n)$. Let assume that $\gcd(f, \partial_xf) = 1$, if it is not the case, we have an obvious factor. Let $V$ be the vector space $$ \left\{ (A,B)\in k[x,y]^2 \ \middle| \ \deg A \leq (m-1,n),\ \deg B \leq (m,n-1) \right \},$$ and $E$ the subspace of all $(A,B)$ in $V$ such that $$ \frac{\partial}{\partial y}\left(\frac A f\right) = \frac{\partial}{\partial x}\left(\frac B f\right).$$

There is an obvious element in $E$: the couple $\left(\partial_x f,\partial_y f\right)$. More interestingly, if $g$ is a polynomial which divides $f$, then the couple $\left(\frac fg \partial_x g, \frac fg \partial_y g\right)$ is in $E$ as well. If $f = gh$, you can check that $$ \left(\partial_x f,\partial_y f\right) = \left(\tfrac fg \partial_x g, \tfrac fg \partial_y g\right) + \left(\tfrac fh \partial_x h, \tfrac fh \partial_y h\right).$$

Now, the following fact should not surprise you : the dimension of E over $k$ is exactly the number of irreducible factors of $f$ over the algebraic closure of $k$.

So just with linear algebra over the rational numbers, I can prove that $xy+x^6+y^6$, as well as the two other polynomials you gave, are irreducible just by computing the rank of a matrix.

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For the sake of completeness, here is a little piece of Maple code to perform the computation.

f := x*y+x^6+y^6:

# A and B are generic elements of V.
A := add(add(a[i,j]*x^i*y^j,i=0..degree(f,x)-1),j=0..degree(f,y)):
B := add(add(b[i,j]*x^i*y^j,i=0..degree(f,x)),j=0..degree(f,y)-1):

# This is the system of equation defining E.
collect(numer(diff(A/f,y)-diff(B/f,x)),[x,y],distributed):
eqs := [coeffs(%,[x,y])]:

# We compute a basis a generic element of E.
sol:=solve(eqs, indets(eqs)): 

# The number of free variables in the generic
# element gives the dimension of E.
nops(indets(subs(sol, indets(eqs))));

This code give you the number of irreducible factors of $f$ !

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• Picard, É. and Simart, G. (1906). Théorie des fonctions algébriques de deux variables indépendantes II
• Ruppert, W. M. (1999). Reducibility of polynomials $f(x,y)$ modulo $p$, J. Number Theory, 77(1), 62--70
• Gao, S. (2003). Factoring multivariate polynomials via partial differential equations, Math. Comp., 72(242), 801--822 (electronic)

This is a brute force outline to show that $xy - x^6 - y^6$ is irreducible. I am not sure what one can do in general.

Assume that it is reducible, being the product of $h(x,y)g(x,y)$, with $\deg h \ge \deg g$. We adopt the notation that $h_i$ (resp $g_i$) represents the degree $i$ part of $h$ (resp $g$).

By looking at the lowest degree term, we see that $h,g$ either starts with $(xy,1)$, $(1,xy)$, or $(x,y)$, $(y,x)$.

Case 1: $\deg g = 1$, then $g$ must be $x$ or $y$, since this is the degree 1 term for $g$. Impossible since $xy$ does not divide $x^6 + y^6$.

Case 2: $\deg g = 2$. $g$ can't be $xy$ by the same divisibility issue.

If $g$ starts with $x$ or $y$, check that the degree 5 term in the product $h(x,y)g(x,y)$ should be nonzero. But it is zero for $xy-x^6-y^6$.

If $g$ starts with 1. Check that the degree 1 term for $g$ is zero by looking at the degree 5 term in the product. Then look at the degree 4 term:

$$0 = h_2g_2 + h_3g_1 + h_4g_0 = xy g_2 + h_4$$

Degree 6 term says $h_4g_2 = -(x^6+y^6)$. This implies that $$h_4 = -xyg_2 \implies -h_4g_2=xyg_2^2 \iff x^6+y^6 = xyg_2^2,$$ But $xy$ does not divide $x^6+y^6$, contradiction.

Case 3: $\deg g = 3$. If $g$ starts with $xy$, look at degree 5 term, contradiction.

If $g$ starts with $x$ or $y$, WLOG let $g$ starts with $x$. Degree 3 term says: $$0 = yg_2 + xh_2$$ Unique factorization forces $y|h_2$ and $x|g_2$. Now look at degree 4 term to conclude that $y|h_3$. But then $y | h$, so $y|x^6+y^6$ which is absurd.

If $g$ starts with 1, look at degree 3 term to conclude that $xy | h_3$. But then $xy | h$, so $xy | x^6+y^6$ which is absurd.

Thus $xy - x^6 - y^6$ is irreducible.