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I don't think that there are any really easy examples. In the famous paper of Beauville, Colliot-Thélène, Sansuc and Swinnerton-Dyer "Variétés stablement rationnelles non rationnelles" they construct surfaces $S$ over $\mathbb Q$ that are not rational, but such that the products $S \times \mathbb P^3$ are rational. You get an example by taking $K$ to be a purely transcendental extension of the function field of $S$ of transcendence degree $d$, and a purely transcendental extension of $\mathbb Q$ of transcendence degree $d+2$, for some $d$ between $0$ and $3$ (I don't know the correct value of $d$).

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I don't think that there any really easy examples. In the famous paper of Beauville, Colliot-Thélène, Sansuc and Swinnerton-Dyer "Variétés stablement rationnelles non rationnelles" they construct surfaces $S$ over $\mathbb Q$ that are not rational, but such that the products $S \times \mathbb P^3$ are rational. You get an example by taking $K$ to be a purely transcendental extension of the function field of $S$ of transcendence degree $d$, and a purely transcendental extension of $\mathbb Q$ of transcendence degree $d+2$, for some $d$ between $0$ and $3$ (I don't know the correct value of $d$).
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