By James Murphy

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Is we we 45 get a resolvent of the first order. get a resolvent of zero order. we let HI we let u If If = =y y% , L , CONSTRUCTION OF IRREDUCIBLE SYSTEMS We 32. shall establish a result a converse of the result of which to is, some A Let be an algebraically irreducible form in w, effectively involving w. Let (44) yi where the Pi and : = --, i are forms in MI, Q extent, 29. , uq \ MI, = w , uq ; ,#, 1, and where Q does not vanish for every solution in the general solution Let MI, ---lUg] w be any solution in the general of A.

What is irreducible. 25, M,-, we can show y< alone. that contains The system 2 composed and the zero form is a closed irreducible more, the theory of resolvents shows that ALGEBRAIC DIFFERENTIAL EQUATIONS 46 yi, yn can be obtained have thus a theoretical process for con- every closed irreducible system in in this We way. , structing all closed irreducible systems. IRREDUCIBILITY AND THE OPEN REGION 33- The question might be raised whether 2, irrecan be reducible in & as to for the open region 21, ducible in 21 some open region 2li in 21.

Let 2 be a non-trivial closed system in y lf yn There may be some y, say yj, such that no non-zero form of 2 involves only yj that is, every form in which yj appears If effectively also involves effectively some yu with k^j. there exist such unknowns yj, let us pick one of them, arbitrarily, and call it U L There may be a y, distinct from u v such that no non-zero , * ; . , * In Chapter VI, we determine which solutions of (5) other than the regular ones are solutions of 2. ALGEBRAIC DIFFERENTIAL EQUATIONS 26 form of such ijj 2 let involves only u and the new y.

### Differential Galois theory by James Murphy

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