Under Abel's guidance, the prevailing obscurities of analysis began to be cleared, new fields were entered upon and the study of functions so advanced as to provide mathematicians with numerous ramifications along which progress could be made.
His works, the greater part of which originally appeared in Crelle's Journal, were edited by Bernt Michael Holmboe and published in 1839 by the Norwegian government, and a more complete edition by Ludwig Sylow and Sophus Lie was published in 1881.
The adjective "abelian", derived from his name, has become so commonplace in mathematical writing that it is conventionally spelled with a lower-case initial "a" (e.g., abelian group, abelian category, and abelian variety).

On 6 April 1929, four Norwegian stamps were issued for the centenary of Abel's death.
His portrait appears on the 500-kroner banknote (version V) issued during 1978–1985.
On 5 June 2002, four Norwegian stamps were issued in honour of Abel two months before the bicentenary of his birth.
There is also a 20-kroner coin issued by Norway in his honour.
A statue of Abel stands in Oslo, and crater Abel on the Moon was named after him.
In 2002, the Abel Prize was established in his memory.

Mathematician Felix Klein wrote about Abel:
But I would not like to part from this ideal type of researcher, such as has seldom appeared in the history of mathematics, without evoking a figure from another sphere who, in spite of his totally different field, still seems related.
Thus, although Abel shared with many mathematicians a complete lack of musical talent, I will not sound absurd if I compare his kind of productivity and his personality with Mozart's.
Thus one might erect a monument to this divinely inspired mathematician like the one to Mozart in Vienna: simple and unassuming he stands there listening, while graceful angels float about, playfully bringing him inspiration from another world.

Instead, I must mention the very different type of memorial that was in fact erected to Abel in Christiania and which must greatly disappoint anyone familiar with his nature.
On a towering, steep block of granite a youthful athlete of the Byronic type steps over two greyish sacrificial victims, his direction toward the heavens.
If needed be, one might take the hero to be a symbol of the human spirit, but one ponders the deeper significance of the two monsters in vain.
Are they the conquered quintic equations or elliptic functions?
Or the sorrows and cares of his everyday life?
The pedestal of the monument bears, in immense letters, the inscription ABEL.
Under Abel's guidance, the prevailing obscurities of analysis began to be cleared, new fields were entered upon and the study of functions so advanced as to provide mathematicians with numerous ramifications along which progress could be made. His works, the greater part of which originally appeared in Crelle's Journal, were edited by Bernt Michael Holmboe and published in 1839 by the Norwegian government, and a more complete edition by Ludwig Sylow and Sophus Lie was published in 1881. The adjective "abelian", derived from his name, has become so commonplace in mathematical writing that it is conventionally spelled with a lower-case initial "a" (e.g., abelian group, abelian category, and abelian variety).
On 6 April 1929, four Norwegian stamps were issued for the centenary of Abel's death. His portrait appears on the 500-kroner banknote (version V) issued during 1978–1985. On 5 June 2002, four Norwegian stamps were issued in honour of Abel two months before the bicentenary of his birth. There is also a 20-kroner coin issued by Norway in his honour. A statue of Abel stands in Oslo, and crater Abel on the Moon was named after him. In 2002, the Abel Prize was established in his memory.
Mathematician Felix Klein wrote about Abel:
But I would not like to part from this ideal type of researcher, such as has seldom appeared in the history of mathematics, without evoking a figure from another sphere who, in spite of his totally different field, still seems related. Thus, although Abel shared with many mathematicians a complete lack of musical talent, I will not sound absurd if I compare his kind of productivity and his personality with Mozart's. Thus one might erect a monument to this divinely inspired mathematician like the one to Mozart in Vienna: simple and unassuming he stands there listening, while graceful angels float about, playfully bringing him inspiration from another world.
Instead, I must mention the very different type of memorial that was in fact erected to Abel in Christiania and which must greatly disappoint anyone familiar with his nature. On a towering, steep block of granite a youthful athlete of the Byronic type steps over two greyish sacrificial victims, his direction toward the heavens. If needed be, one might take the hero to be a symbol of the human spirit, but one ponders the deeper significance of the two monsters in vain. Are they the conquered quintic equations or elliptic functions? Or the sorrows and cares of his everyday life? The pedestal of the monument bears, in immense letters, the inscription ABEL.
Niels Henrik Abel