DVI doesn't transfer audio, only video. If you're using DVI for video you'll need to use the audio jack on the NUC for your audio output.
In the german section of Wikipedia, one can find: "Durch die Verwendung nicht dafür vorgesehener Pins können sogar zusätzlich digitale Audiodaten über den DVI-I-Anschluss übertragen werden" =" By using of pins not dedicated to that aim it is even possible to transmit digital audio data via the DVI-I connection." The monitor in use (with two built-in loudspeakers) was three years connected to a mini PC sending audio data to him.
Possible I can believe, but I don't think it will give you audio by default. Plus, you're not using a straight DVI cable, you're using an adapter, which only receives the video signal.
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Thank you very much to jasonNUC for the information provided above, it is always valuable.
To GerhardS, I hope the information provided previously on this thread was useful for you.
And, just to confirm that, if you need to get sound using HDMI, you need and HDK straight cable.
Any questions, please let me know.