NOVELTY - A multi-branched polymer has structure (I).    USE - For adhesive agent, binder for electrode preparation and solid polymer electrolyte (all claimed) used for solid polymer electrolyte battery. As resist material for solid polymer electrolyte battery used as power supply for machines, such as notebook personal computer, electric shaver and liquid crystal television, capacitor, sensor, condenser, electrochemical materials, such as electrochemical element and optoelectric transducer, inclusion material, function material for electrical appliances and industrial apparatus.    ADVANTAGE - The multi-branched polymer has controlled molecular weight, and enables manufacture of solid polymer electrolyte having excellent thermal characteristics, mechanical strength and ion electron conductivity. The multi-branched polymer is produced simply and effectively.    DETAILED DESCRIPTION - A multi-branched polymer has structure of formula (I).    A=organic group having branched chain of 3 or more;    Xa=coupling group containing an atom of periodic table 14-16 groups;    Y=functional group of structure having an active halogen atom;    m=integer of 1 to number of branches bonded to A;    l=number of branches bonded to A;    n=integer of 0 or more;    Q=branch having repeating unit derived by polymerizable unsaturated bond;and    Ra=organic group inert to polymerization.    When m is 2 or more, the groups expressed with formula: (Xa)n-Y-Q (where Q is branch having repeating unit derived by polymerizable unsaturated bond) is same or different. When n is 2 or more, Xa is same or different. When (l-m) is 2 or more, Ra is same or different. INDEPENDENT CLAIMS are included for the following: (i) solid polymer electrolyte having core portion with three or more branched chain, and arm portion having repeating unit of formula (a1); (ii) adhesive agent having core portion and arm portion having repeating unit (a1); and (iii) binder for electrode preparation having core portion and arm portion having repeating unit (a1).    R100,R200,R300=H or 1-10C hydrocarbon;    R4a,R4b=H or methyl;    R500=H, hydrocarbon, acyl or silyl;and    s=2-100.    R100 and R300 are optionally bonded to form a ring.    DESCRIPTION OF DRAWING(S) - The figure shows the graph showing change of electric conductivity of the adhesive agent. (Drawing includes non-English language text).