An introduction to the modern theory of equations, by Florian Cajori.

GALOIS RESOLVENT BY ADJUNCTION 1 175 For suppose s,.ti and s,.t yielded the same value for M; that is, suppose iis = M 1operated upon by s,.t, and M, = M operated upon by s,.t, then, operating with (s,.ti)-1 upon Mi would give MS = M operated upon by (s,tk)(s,ti) —l That is, (s,.tk)(sti)-l is a substitution contained in the group Q and is equal to, say s,,. If s, _ (s,.t)(s,.t)-1, then, operating with st,, we get stk =sr S t $rtk -- 8mSr'i -- $,Otti? where sm' is a substitution in Q. Since the effects of s,.tk and sM't, upon M2 are the effects of tk and ti alone, it follows that t = ti, which is contrary to supposition. Hence s,.ti and s,.tk must yield different values when applied to M.1. The function S((y) (y — M)(y- MD) *. (y- Mj_) is now seen to be invariant under any substitution of P. The coefficients of y in <(/), obtained by performing the indicated multiplications, are symmetric functions of M11, M1,, MJ_-1 and, therefore, by the definition of iM, functions in 0~ of the roots of f(x) = 0 functions which admit of the substitutions of the Galois group P. Hence these coefficients are numbers in 2 (~ 154). To prove the irreducibility of >(/), assume that O(y) is any function of y in Q, which vanishes for y = M. Then 0(11) = 0. Since O(M) must acdmit all the substitutions of the Galois group (~ 153), we must have 0(l1.i)= 0, where i has any value 0, 1, 2,., (j- 1). Hence 0(y) cannot be of lower degree than the jth. As all the roots 3,M, M, M, of ~(y) = 0 satisfy O(y) = 0, 0(yJ) is divisible by 0(y)). Now, if ~(y) were reducible, one of its factors would vanish for y =M. Since 0(y) may be any algebraic function in 52 which vanishes for y = M, let 0(y) represent this factor. Then it would follow that this factor would be divisible by the whole product 0(y), which is impossible. Hence <S(y) is irreducible.

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Title
An introduction to the modern theory of equations, by Florian Cajori.
Author
Cajori, Florian, 1859-1930.
Canvas
Page 170
Publication
New York,: The Macmillan company,
1904.
Subject terms
Equations, Theory of
Group theory.

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"An introduction to the modern theory of equations, by Florian Cajori." In the digital collection University of Michigan Historical Math Collection. https://name.umdl.umich.edu/abv2146.0001.001. University of Michigan Library Digital Collections. Accessed May 29, 2025.
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