Proof for Kolmogorov complexity is uncomputable using reductions
You can find two different proofs in:

Gregory J. Chaitin, Asat Arslanov, Cristian Calude: Program-size Complexity Computes the Halting Problem. Bulletin of the EATCS 57 (1995)

In Li, Ming, Vitányi, Paul M.B.; An Introduction to Kolmogorov Complexity and Its Applications it is presented as an exercise (with a hint on how to solve it that is credited to P. Gács by W. Gasarch in a personal communication Feb 13, 1992).

** I decided to publish an extended version of it on my blog.