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The answer seems to be yes. I think this was proved by Kuhlmann, Pank, and Roquette, "Immediate and purely wild extensions of valued fields""Immediate and purely wild extensions of valued fields", Manuscripta math. 55 (1986), 39-67. A short proof is given in Efrat's book on valuation theory, p. 203.

The answer seems to be yes. I think this was proved by Kuhlmann, Pank, and Roquette, "Immediate and purely wild extensions of valued fields", Manuscripta math. 55 (1986), 39-67. A short proof is given in Efrat's book on valuation theory, p. 203.

The answer seems to be yes. I think this was proved by Kuhlmann, Pank, and Roquette, "Immediate and purely wild extensions of valued fields", Manuscripta math. 55 (1986), 39-67. A short proof is given in Efrat's book on valuation theory, p. 203.

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The answer seems to be yes. I think this was proved by Kuhlmann, Pank, and Roquette, "Immediate and purely wild extensions of valued fields", Manuscripta math. 55 (1986), 39-67. A short proof is given in Efrat's book on valuation theory, p. 203.