Look first at the colours, then at the additional emblem.
The flag in the original question has three horizontal stripes: black at the top, red in the middle and gold at the bottom. A prominent emblem sits in the centre. This additional emblem distinguishes the flag sought from the plain Federal Flag.
After their founding, the GDR and the Federal Republic initially used the same colours without this distinguishing GDR emblem on the flag. The addition in 1959 made the version shown here clearly identifiable.
The other three answers
The complete appearance matters, not one individual colour.
Prussia
The combination of black-red-gold and the described state emblem belongs to the GDR. It is not a Prussian state flag.
Federal Republic of Germany
The Federal Flag is also black-red-gold, but without the hammer, compass and grain wreath. The federal service flag uses a shield with an eagle and is likewise different from the flag shown.
'Third Reich'
The emblem shown belongs to the post-war GDR. The term “Third Reich” refers instead to Nazi rule and does not match this flag.
Not every black-red-gold flag is the same.
Additional design details in historical pictures can help identify the state and period. The colours alone are insufficient here. It would also be wrong to treat every central emblem as a GDR emblem.
The GDR had existed since 1949. The fact that the emblem appeared on its flag only in 1959 does not change the state’s founding year. Changing a symbol and founding a state are different events.
GDR and state symbols · House of History ↗Black-red-gold + hammer, compass, grain wreath → GDR.
Remember the combination of the three colours and this particular emblem.
Three words for this question
- das Staatswappenstate coat of arms / state emblem
- A prescribed pictorial emblem of a state.
- der Zirkeldrawing compass
- A tool for drawing circles; here, part of the emblem.
- der Ährenkranzwreath of grain
- A wreath of ears of grain surrounding the other symbols here.