We present K-band spectra of rest-frame optical emission lines for 24 star-forming galaxies at z~3.2-3.7 using MOSFIRE on the Keck 1 telescope. Strong rest-frame optical [O III] and Hbeta emission lines were detected in 18 LBGs. The median flux ratio of [O III]5007 to Hbeta is 5.1+/-0.5, a factor of 5-10x higher than in local galaxies with similar stellar masses. The observed Hbeta luminosities are in good agreement with expectations from the estimated star-formation rates, and none of our sources are detected in deep X-ray stacks, ruling out significant contamination by active galactic nuclei. Combining our sample with a variety of LBGs from the literature, including 49 galaxies selected in a very similar manner, we find a high median ratio of [OIII]/Hbeta = 4.8+0.8-1.7. This high ratio seems to be an ubiquitous feature of z~3-4 LBGs, very different from typical local star-forming galaxies at similar stellar masses. The only comparable systems at z~0 are those with similarly high specific star-formation rates, though ~5x lower stellar masses. High specific star-formation rates either result in a much higher ionization parameter or other unusual conditions for the interstellar medium, which result in a much higher [OIII]/Hbeta line ratio. This implies a strong relation between a global property of a galaxy, the specific star-formation rate, and the local conditions of ISM in star-forming regions.
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