If I were to write a song in D minor, could I notate it correctly in F Aeolian? Because the subsequent 4th of the root in F Aeolian would be flat, right? Therefore, D minor and F Aeolian, which in different keys like C Aeolian and A minor are the same, would be different by the B flat? Please correct me if I'm mistaken.
(Natural) Minor and Aeolian are two words describing the same scale, as are Major and Ionian.
D minor/aeolian and F major/ionian share the same key signature: Bb.
A minor/aeolian and C major/ionian share the same key signature: no sharps or flats.
These major/minor (ionian/aeolian) pairs are called relative modes, meaning that they share the same key signatures.
The third note of a minor/aeolian scale is the first note of the corresponding (relative) major/ionian scale. The sixth note of a major/ionian scale is the first note of the corresponding (relative) minor/aeolian scale.
Key and mode are not the same thing.
In a mode (in the modern/current meaning of the word), your tonic (home note, center pitch) and the intervals around it are fixed in rigid positions, and if you play pitches not included in the fixed set, clearly enough to make a harmonic impact, you're not in the mode anymore.
In a key, only your tonic is fixed, and the third above it, when the harmony is at home, resting at the center of balance. In minor keys it's a minor third above tonic, and in major keys it's a major third above tonic (home note). But all the rest of the intervals can and often do change along the way. However, a key has a default set of notes/pitches, which is denoted by a key signature, and in a minor key, the default set of notes/pitches is the same as in the corresponding Aeolian. But in a key, there can be chromatic alterations, but the listener's feeling still remaining in the same key. The chromatic alterations are denoted with temporary accidentals.
If your sense of tonic does not change, i.e. where home is and if home is minor or major, the key does not change. But if the intervals around the tonic change, the mode changes.
You can have the chords Dm, E7, A7, Dm, in a tune that's in the key of D minor. During the E7 chord, the G note is temporarily sharpened to G#, and during the A7 chord, the C note is temporarily sharpened to C#. But you cannot have that in D Aeolian mode. In a mode, there are no chromatic alterations of notes. Inside a mode, you stick to the given intervals around the tonic and do not change them.
If your song is in D minor, chances are that it will use pitches outside D Aeolian, particularly C#, which is the so-called leading tone for D. And then you'll need to write temporary accidentals in addition to the key signature. Some people would even go as far as to say, if you do NOT raise the C to C# anywhere in the song, your song is not properly in D minor, because there's no leading tone.
Furthermore, the key signature of D minor has the notes of D Aeolian, not F Aeolian. F Aeolian has F, G, Ab, Bb, C, Db, Eb.
Edit. The above was apparently not clear enough. I'll try once more, with pictures.
In traditional Western harmony thinking, there are seven scale degrees and a tonal center, tonic. The tonal center is a center of balance - you're supposed to feel when the music's harmony is "at home", resting. (In addition to the tonic being a note, it has a major/minor dimension to it, an expectation of a third being suitable beside the tonic at the resting position.) Each of the scale degrees can be thought to be like a switch that can be in natural, sharp or flat setting. (Let's not worry about double-flat and double-sharp etc. now.)
A key signature specifies settings for the seven scale degrees, but it does not say where the tonal center is. The notes you hear make you feel something about the tonal center. And it is entirely possible to use the same set of seven notes and set the tonal center to any of the notes, in the listener's mind. For some scale degrees it might take more skill than for others, but it's possible. If you feel that D is the tonal center - as you do if your song is, say, in D minor key, then the D note has special meaning to you:
Additionally, you can write temporary accidentals, which change a staff position to something different than what was specified by the key signature. For example, if you started with one flat, i.e. Bb and the rest of the scale degrees in natural, but you play an A7 chord, you'll most likely consider the C note as being switched to sharp position, at least for the duration of the chord. Look what happened to the C note's switch:
You might think something different about other scale degrees if you want, but the sounding pitches don't explicate things about the other notes. If you strongly PLAY a B natural, then the B scale degree's switch in your mind tends to get switched to natural position. You can think that you have harmonic context in your mind, consisting of a set of switches that change between sharp/natural/flat, according to what pitches you hear.
When you're "in a key", you can freely switch the scale degrees to different sharp/flat/natural positions, as long as you don't mess with the feeling of tonal center. If the tonal center moves, then the key changed. That's called a modulation.
Ok, what's a mode then? To stay in a mode, you are NOT free to change scale degrees, and you are NOT free to move the tonal center either. Be careful what you play, if you want to stay in the mode. The swithes should be like they had been superglued in place:
If you DO mess with any of the settings, then you break the mode and you're no longer in the same mode. Let's say you play a C# note. Not D Aeolian anymore.
If I were to write a song in D minor, could I notate it correctly in F Aeolian?
I think you meant
F Ionian or
F major, and you mean the key signatures involved.
So, could a song in
D minor, which would normally have a key signature of one flat, be written with a
F major key signature of one flat? Yes, both key signatures use one flat.
Is D minor also F Aeolian?
Again, assuming you meant
F Ionian: no, they are not the same.
They each have different tonics and modes.
D minor has a tonic of
D and a mode of minor.
F Ionian has a tonic of
F and a mode of Ionian (major.)
Additionally, the dominant seventh chord of each will be different.
D minor has a dominant of
F major has a dominant of
Keys are determined by tonic/dominant harmony, not key signatures.
Some types of modal music - like folk and Renaissance music - work in a similar way but may use other chords as a dominant such as
iv rather than
V. But the point still remains: the tonality (for lack of a better term, the modal "key") will be determined by the interaction of tonic and modal dominant chords.
One other point. You should understand and make clear whether you mean music that is actually in a minor key or a modal style of harmony in aeolian mode. Using "in minor" and "in aeolian" as interchangeable terms is wrong and confusing. It doesn't have immediate impact on your question, but it makes a jumble of the terminology and concepts of tonality. That could make it hard to understand your intentions if the topic is harmony and tonality.
No, D minor is the name of a key which is highly compatible with the D Aeolian scale.
"F Aeolian" is not a term which means "the Aeolian mode relative to F major". However, yes, D Aeolian is a mode of the F major scale, or F Ionian.
"F Aeolian" has the simplest possible meaning: the Aeolian mode starting on the F root note. Music emphasizing F Aeolian would be most likely be identified as being in the F minor key.