Let p be an odd prime number, and a be an integer divisible by p exactly once. We prove that the Galois group G of the trinomial X^{p^{2}}+aX+a over the field Q of rational number is either the full symmetric group S_{p^{2}} or G lies between AGL(1,p^{2}) and AGL(2,p)$. Furthermore, we establish conditions when G is S_{p^{2}}.
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