We present Keck/MOSFIRE spectra of the diagnostic nebular emission lines [OIII]5007,4959, [OII]3727, and H-beta for a sample of 15 redshift z=3.1-3.7 Ly-alpha emitters (LAEs) and Lyman break galaxies (LBGs). In conjunction with spectra from other surveys, we confirm earlier indications that LAEs have a much higher [OIII]/[OII] line ratio than is seen in similar redshift LBGs. By comparing their distributions on a [OIII]/[OII] versus R23 diagram, we demonstrate that this difference cannot arise solely because of their lower metallicities but most likely is due to a harder ionizing spectrum. Using measures of H-beta and recombination theory, we demonstrate, for a subset of our LAEs, that xi_ion - the number of Lyman continuum photons per UV luminosity - is indeed 0.2-0.5 dex larger than for typical LBGs at similar redshifts. Using photoionization models we estimate the effect this would have on both [OIII]/[OII] and R23 and conclude such a hard spectrum can only partially explain such intense line emission. The additional possibility is that such a large [OIII]/[OII] ratio is in part due to density rather than ionization bound nebular regions, which would imply a high escape fraction of ionizing photons. We discuss how further observations could confirm this possibility. Clearly LAEs with intense [OIII] emission represent a promising analog of those z>7 sources with similarly strong lines which are thought to be an important contributor to cosmic reionization.
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