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Let $f\in A[x][y]$ where $A$ is a UFD and assume that $f$ is monic i.e that it can be written as $$ f=a_0+a_1 y+\ldots a_{n-1}y^{n-1}+y^n, $$ with $a_i\in A[x]$. Assume that there exists an irreducible polynomial $g\in A[x]$ such that $g|a_i$ for $i=0\ldots n-1$ and $g^2\nmid a_0$ then by Eisenstein's criterion $f$ is irreducible over $A[x]$ and therefore irreducible in $A[x,y]$. I've used the fact that $A$ was a UFD only to make sure that $gA[x]$ in $A[x]$ is a prime ideal. (Of course Eisenstein criterion is a very special case which fits within the method of the Newton polygon).

Using this idea and induction it is easy to see that polynomials like $$ (*)\;\;\;\; x_1^{n_1}+x_2^{n_2}+\ldots x_r^{n_r}\in \mathbf{C}[x_1,x_2,\ldots x_r], $$ are irreducible whenever $n_i\geq 1$ and $r\geq 3$, since the polynomial $x_1^{n_1}+x_2^{n_2}$ has always a multiplicity one irreducible divisor.

More generally there is the so called Eherenfeucht criterion which says that $$ (**) \;\;\;\; f_1(x_1)+f_2(x_2)+\ldots f_r(x_r)\in\mathbf{C}[x_1,\ldots,x_r], $$ is always irreducible if $deg(f_i)\geq 1$ and $r\geq 3$. In the case where $r=2$ it is still irreducible if one has $(deg(f_1),deg(f_2))=1$.

Note that the polynomials in $(\star)$ are a very special case of polynomials in $(\star\star)$. A nice proof of this criterion may be found in a paper of Tverberg entitled "A remark on Ehrenfeucht's crieterion for irreducibility of polynomials". Unfortunately, if you have a polynomial with mixed monomials then this criterion does not apply.

Let $f\in A[x][y]$ where $A$ is a UFD and assume that $f$ is monic, i.e that it can be written as $$ f=a_0+a_1 y+\ldots a_{n-1}y^{n-1}+y^n, $$ with $a_i\in A[x]$. Assume that there exists an irreducible polynomial $g\in A[x]$ such that $g|a_i$ for $i=0\ldots n-1$ and $g^2\nmid a_0$ then by Eisenstein's criterion $f$ is irreducible over $A[x]$ and therefore irreducible in $A[x,y]$. I've used the fact that $A$ was a UFD only to make sure that $gA[x]$ in $A[x]$ is a prime ideal. (Of course Eisenstein criterion is a very special case which fits within the method of the Newton polygon).

Using this idea and induction it is easy to see that polynomials like $$ (*)\;\;\;\; x_1^{n_1}+x_2^{n_2}+\ldots x_r^{n_r}\in \mathbf{C}[x_1,x_2,\ldots x_r], $$ are irreducible whenever $n_i\geq 1$ and $r\geq 3$, since the polynomial $x_1^{n_1}+x_2^{n_2}$ has always a multiplicity one irreducible divisor.

More generally there is the so called Ehrenfeucht criterion which says that $$ (**) \;\;\;\; f_1(x_1)+f_2(x_2)+\ldots f_r(x_r)\in\mathbf{C}[x_1,\ldots,x_r], $$ is always irreducible if $\deg(f_i)\geq 1$ and $r\geq 3$. In the case where $r=2$ it is still irreducible if one has $(\deg(f_1),\deg(f_2))=1$.

Note that the polynomials in $(\star)$ are a very special case of polynomials in $(\star\star)$. A nice proof of this criterion may be found in a paper of Tverberg entitled "A remark on Ehrenfeucht's criterion for irreducibility of polynomials". Unfortunately, if you have a polynomial with mixed monomials then this criterion does not apply.

Let $f\in A[x][y]$ where $A$ is a UFD and assume that $f$ is monic i.e that it can be written as $$ f=a_0+a_1 y+\ldots a_{n-1}y^{n-1}+y^n, $$ with $a_i\in A[x]$. Assume that there exists an irreducible polynomial $g\in A[x]$ such that $g|a_i$ for $i=0\ldots n-1$ and $g^2\nmid a_0$ then by Eisenstein's criterion $f$ is irreducible over $A[x]$ and therefore irreducible in $A[x,y]$. I've used the fact that $A$ was a UFD only to make sure that $gA[x]$ in $A[x]$ is a prime ideal. (Of course Eisenstein criterion is a very special of the method of the Newton polygon).

Using this idea and induction it is easy to see that polynomials like $$ (*)\;\;\;\; x_1^{n_1}+x_2^{n_2}+\ldots x_r^{n_r}\in \mathbf{C}[x_1,x_2,\ldots x_r], $$ are irreducible whenever $n_i\geq 1$ and $r\geq 3$, since the polynomial $x_1^{n_1}+x_2^{n_2}$ has always a multiplicity one irreducible divisor.

More generally there is the so called Eherenfeucht criterion which says that $$ (**) \;\;\;\; f_1(x_1)+f_2(x_2)+\ldots f_r(x_r)\in\mathbf{C}[x_1,\ldots,x_r], $$ is always irreducible if $deg(f_i)\geq 1$ and $r\geq 3$. In the case where $r=2$ it is still irreducible if one has $(deg(f_1),deg(f_2))=1$.

Note that the polynomials in $(\star)$ are a very special case of polynomials in $(\star\star)$. A nice proof of this criterion may be found in a paper of Tverberg entitled "A remark on Ehrenfeucht's crieterion for irreducibility of polynomials". Unfortunately, if you have a polynomial with mixed monomials then this criterion does not apply.

Let $f\in A[x][y]$ where $A$ is a UFD and assume that $f$ is monic i.e that it can be written as $$ f=a_0+a_1 y+\ldots a_{n-1}y^{n-1}+y^n, $$ with $a_i\in A[x]$. Assume that there exists an irreducible polynomial $g\in A[x]$ such that $g|a_i$ for $i=0\ldots n-1$ and $g^2\nmid a_0$ then by Eisenstein's criterion $f$ is irreducible over $A[x]$ and therefore irreducible in $A[x,y]$. I've used the fact that $A$ was a UFD only to make sure that $gA[x]$ in $A[x]$ is a prime ideal. (Of course Eisenstein criterion is a very special case which fits within the method of the Newton polygon).

Using this idea and induction it is easy to see that polynomials like $$ (*)\;\;\;\; x_1^{n_1}+x_2^{n_2}+\ldots x_r^{n_r}\in \mathbf{C}[x_1,x_2,\ldots x_r], $$ are irreducible whenever $n_i\geq 1$ and $r\geq 3$, since the polynomial $x_1^{n_1}+x_2^{n_2}$ has always a multiplicity one irreducible divisor.

More generally there is the so called Eherenfeucht criterion which says that $$ (**) \;\;\;\; f_1(x_1)+f_2(x_2)+\ldots f_r(x_r)\in\mathbf{C}[x_1,\ldots,x_r], $$ is always irreducible if $deg(f_i)\geq 1$ and $r\geq 3$. In the case where $r=2$ it is still irreducible if one has $(deg(f_1),deg(f_2))=1$.

Note that the polynomials in $(\star)$ are a very special case of polynomials in $(\star\star)$. A nice proof of this criterion may be found in a paper of Tverberg entitled "A remark on Ehrenfeucht's crieterion for irreducibility of polynomials". Unfortunately, if you have a polynomial with mixed monomials then this criterion does not apply.

Let $f\in A[x][y]$ where $A$ is a UFD and assume that $f$ is monic i.e that it can be written as $$ f=a_0+a_1 y+\ldots a_{n-1}y^{n-1}+y^n, $$ with $a_i\in A[x]$. Assume that there exists an irreducible polynomial $g\in A[x]$ such that $g|a_i$ for $i=0\ldots n-1$ and $g^2\nmid a_0$ then by Eisenstein's criterion $f$ is irreducible over $A[x]$ and therefore irreducible in $A[x,y]$. I've used the fact that $A$ was a UFD only to make sure that $gA[x]$ in $A[x]$ is a prime ideal.

Using this idea and induction it is easy to see that polynomials like $$ (*)\;\;\;\; x_1^{n_1}+x_2^{n_2}+\ldots x_r^{n_r}\in \mathbf{C}[x_1,x_2,\ldots x_r], $$ are irreducible whenever $n_i\geq 1$ and $r\geq 3$, since the polynomial $x_1^{n_1}+x_2^{n_2}$ has always a multiplicity one irreducible divisor.

More generally there is the so called Eherenfeucht criterion which says that $$ (**) \;\;\;\; f_1(x_1)+f_2(x_2)+\ldots f_r(x_r)\in\mathbf{C}[x_1,\ldots,x_r], $$ is always irreducible if $deg(f_i)\geq 1$ and $r\geq 3$. In the case where $r=2$ it is still irreducible if one has $(deg(f_1),deg(f_2))=1$.

Note that the polynomials in $(\star)$ are a very special case of polynomials in $(\star\star)$. A nice proof of this criterion may be found in a paper of Tverberg entitled "A remark on Ehrenfeucht's crieterion for irreducibility of polynomials". Unfortunately, if you have a polynomial with mixed monomials then this criterion does not apply.

Let $f\in A[x][y]$ where $A$ is a UFD and assume that $f$ is monic i.e that it can be written as $$ f=a_0+a_1 y+\ldots a_{n-1}y^{n-1}+y^n, $$ with $a_i\in A[x]$. Assume that there exists an irreducible polynomial $g\in A[x]$ such that $g|a_i$ for $i=0\ldots n-1$ and $g^2\nmid a_0$ then by Eisenstein's criterion $f$ is irreducible over $A[x]$ and therefore irreducible in $A[x,y]$. I've used the fact that $A$ was a UFD only to make sure that $gA[x]$ in $A[x]$ is a prime ideal. (Of course Eisenstein criterion is a very special of the method of the Newton polygon).

Using this idea and induction it is easy to see that polynomials like $$ (*)\;\;\;\; x_1^{n_1}+x_2^{n_2}+\ldots x_r^{n_r}\in \mathbf{C}[x_1,x_2,\ldots x_r], $$ are irreducible whenever $n_i\geq 1$ and $r\geq 3$, since the polynomial $x_1^{n_1}+x_2^{n_2}$ has always a multiplicity one irreducible divisor.

More generally there is the so called Eherenfeucht criterion which says that $$ (**) \;\;\;\; f_1(x_1)+f_2(x_2)+\ldots f_r(x_r)\in\mathbf{C}[x_1,\ldots,x_r], $$ is always irreducible if $deg(f_i)\geq 1$ and $r\geq 3$. In the case where $r=2$ it is still irreducible if one has $(deg(f_1),deg(f_2))=1$.

Note that the polynomials in $(\star)$ are a very special case of polynomials in $(\star\star)$. A nice proof of this criterion may be found in a paper of Tverberg entitled "A remark on Ehrenfeucht's crieterion for irreducibility of polynomials". Unfortunately, if you have a polynomial with mixed monomials then this criterion does not apply.