Significance
Artin's reciprocity law implies a description of the abelianization of the absolute Galois group of a global field K which is based on the local–global principle and the use of the Frobenius elements. Together with the Takagi existence theorem, it is used to describe the abelian extensions of K in terms of arithmetic of K and to understand the behavior of the nonarchimedean places in them. Therefore, the Artin reciprocity law can be interpreted as one of the main theorems of global class field theory. It can be used to prove that Artin L-functions are meromorphic and for the proof of the Chebotarev density theorem.
Read more about this topic: Artin Reciprocity Law
Famous quotes containing the word significance:
“It is necessary not to be Christian to appreciate the beauty and significance of the life of Christ.”
—Henry David Thoreau (18171862)
“To grasp the full significance of life is the actors duty, to interpret it is his problem, and to express it his dedication.”
—Marlon Brando (b. 1924)
“Politics is not an end, but a means. It is not a product, but a process. It is the art of government. Like other values it has its counterfeits. So much emphasis has been placed upon the false that the significance of the true has been obscured and politics has come to convey the meaning of crafty and cunning selfishness, instead of candid and sincere service.”
—Calvin Coolidge (18721933)