Some tones of a chord can be omitted while other tones are essential for defining the chord.
In jazz there is a rootless chord voicing, but don't confuse that with a true root omission. It's typical that a bass part will play the root while some other instrument play rootless chord voicings for part of the accompaniment. Also, shells are another kind of jazz accompaniment chord that are incomplete, but the other musical parts often supply additional tones which fill in the harmony.
Have said that about root omission, there is one common chord that is sometimes described theoretically as a root-omitted chord: the diminished seventh chord. If you take a dominant ninth chord in the minor mode and then omit it's root, the remaining chord is a diminished seventh chord built on the leading tone. As the chord is theoretically derived from a dominant chord, the diminished seventh leading tone chord is then considered a type of dominant chord.
Otherwise it's not normal to omit the root from a chord.
The third of chord is more or less required. It give the basic chord quality of major or minor.
The fifth of most chords will be a perfect fifth. Diminished fifths would typically indicate a chord built on a leading tone or a subdominant
iio chord in minor. In jazz diminished or augmented fifths are typical of altered dominant chords. The fifth is often omitted unless a diminished or augmented fifth is needed to make clear diminished or altered chords.
The seventh is required for seventh chords.
Extensions beyond the seventh - ninth, eleventh, and thirteen - usually need the seventh to be present along with the extensions.
So, omission of chord tones depends greatly on the type of chord that will be implied. A reasonable list of essential tones is:
- triads need the root and third
- seventh chords need the root, third and seventh
- extended chords need the root, third, seventh and extensions
...The idea being that if I need to I can omit the non-core tones (the 1st and 5th) and build the harmony around just the defining third and seventh, adding extensions to the 3rd and 7th to achieve the same overall effect...
With dominant chord, yes. This is sort of like the omitting the root on a dominant ninth to make a leading tone diminished seventh chord. When a dominant's third and seventh (solfege
FA) resolve to the next chord's root and third (analogous to
MI) it's clearly understood as dominant harmony. This works because of the characteristic half-step motions of the leading tone and the subdominant degrees, and because the progression is ubiquitous in tonal music.
You might be more careful with other chords. If you omit the root from a
vi7 chord (
Am7 in the key of
C major) you risk have the remaining tones (
C E G) sounding like the plain tonic triad. That doesn't necessarily mean the harmony will be bad. Decide on what is important. If you really want the sound of the minor seventh chord, include the root and seventh. Generally speaking, don't omit essential chord tones.