How can one show (hopefully in an elementary manner) that there exist irreducible polynomials of arbitrary degree and number of variables over arbitrary field? Does ∀n,d∈N∀ field F exist an irreducible f∈F[x1,⋯,xn] of degree d?

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How can one show (hopefully in an elementary manner) that there exist irreducible polynomials of arbitrary degree and number of variables over arbitrary field?  Does ∀n,d∈N∀ field F exist an irreducible f∈F[x1,⋯,xn] of degree d?

Mary Goodson · Accepted Answer

Step 1  This is false. Any irreducible polynomial in one variable over an algebraically closed field (such as C) has degree 1.  For n≥2, the answer is yes: just take the irreducible polynomial y−xd  Proof: If  x1−x2d=f(x)g(x)=(∑aαaxa)(∑bβbxb)=∑a,bαaβbxa+b,  then fg would include a monomial with x1x2 a contradiction.

How can one show (hopefully in an elementary manner) that

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