Zariski cotangent space



In Mamford’s “The Red Book of Varieties and Schemes” in paragraph 3.4 asserts that in closed point Zariski cotangent space m_x/m_x^2 isomorphic \Omega_{X/k}_x \otimes k(x). Is it right for non algebraically closed field k? I think it is right in rational points (k(x)=k) only.
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