TECHNOLOGY FOCUS - ORGANIC CHEMISTRY - Preferred Components: The oxygen-free atmosphere is an atmosphere purged with argon gas, an atmosphere purged with nitrogen gas, or the inside of a vacuum vessel. The radiation for irradiation is a gamma -ray or an electron beam.    TECHNOLOGY FOCUS - POLYMERS - Preferred Components: The graft polymerization is radiation graft polymerization. The polymer base material mainly made of the amide resin comprises an amide polymer, an amide copolymer, or a composite material mainly made of at least one of the polymer. The anionic monomer does not contain other monomers except a crosslinking agent. The polymer base material is: a fibrous base material which is fibers, yarn, hollow fibers, rope, or their composite materials, and the anionic polymer ion-exchange material is a fibrous material; a sheet-form base material which is a film, membrane, woven fabric, nonwoven fabric, or their laminated materials, and the anionic polymer ion-exchange material is a sheet-form material; or a three-dimensional shaped base material which is pellets, column, porous material, foamed material, or their composite materials, and the anionic polymer ion-exchange material is a three-dimensional shaped material. The graft polymer chain of the anionic monomer is present at 20-100 wt.%, based on the mass of the polymer base material. The amide resin has a principal chain of formula (I) or (II).x = 0-1;m = 3-12; andn = not defined.Preferred Method: The method comprises: placing a polymer base material mainly made of an amide resin having both an aromatic structure and an aliphatic chain in its principal chain in an apparatus for isolating the inside from an external environment to maintain an oxygen-free atmosphere and allowing the polymer base material to stay in the apparatus for a predetermined period of time or more so that the polymer base material contains no oxygen; irradiating the polymer base material, which is mainly made of an amide resin having both an aromatic structure and an aliphatic chain in its principal chain and which contains no oxygen, with radiation in an oxygen-free atmosphere; dissolving an anionic monomer having an aromatic structure and a quaternary ammonium structure in a polar solvent deaerated for oxygen gas in an oxygen-free atmosphere; conducting a graft polymerization in which the polymer base material irradiated with radiation is immersed in the solution of the anionic monomer in the oxygen-free atmosphere; and maintaining the irradiated polymer base material in a state of being immersed in the solution of the anionic monomer in the oxygen-free atmosphere to effect the graft polymerization. EXAMPLE - Polymer membrane base material comprised of a semi-aromatic polyamide obtained by condensation polymerization of meta-xylylenediamine and adipic acid was cut, and placed in a separable glass vessel having a cock and deaerated, and then argon gas was filled in the vessel so that the membrane base material did not contain any oxygen. The resultant membrane base material was irradiated with a gamma -ray at room temperature at an exposure dose of 60 kGy. The glass vessel was placed in 80 ml of a methanol solution prepared by dissolving vinylbenzyl trimethylammonium chloride in methanol, while oxygen gas was deaerated in an oxygen-free atmosphere, and the membrane base material was immersed in the methanol solution. The glass vessel was purged with argon gas and then closed, and a reaction was effected at 60 degrees C for a grafting time of 24 hours to obtain anionic polymer ion-exchange membrane.