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Let \(\alpha \) be a non-zero algebraic unit which is not a root of unity and K be a number field of degree d over \(\mathbb {Q}\) . In this paper, we prove the following: Let \(\mathcal {P}\) be a prime ideal of \(\mathcal {O}_K\) which lies above a rational odd prime p such that $$\begin{aligned} \mathcal {P}\mathcal {O}_{K(\alpha )}=\mathscr {P}_1^{e_1}\mathscr {P}_2^{e_2}\cdots \mathscr {P}_g^{e_g}, \end{aligned}$$ where \(\max _{1\le i\le g}\{e_i\}\le p\) and \(e_1+\cdots +e_g=[K(\alpha ):K].\) Then \(h(\alpha )\ge c,\) where \(c>0\) is an effectively computable constant depending only on p and \([K:\mathbb {Q}]=d.\) This generalizes a result of Petsche. Also, we prove the following: Let \(\mathcal {P}\) be a prime ideal of \(\mathcal {O}_K\) which lies above 2 such that $$\begin{aligned} \mathcal {P}\mathcal {O}_{K(\alpha )}=\mathscr {P}_1^{e_1}\mathscr {P}_2^{e_2}\cdots \mathscr {P}_r^{e_r}, \end{aligned}$$ where \(e_1+\cdots +e_r=[K(\alpha ):K].\) Then \(M(\alpha )\ge C(K),\) where \(C(K)>1\) is a constant which depends only on \([K:\mathbb {Q}]=d.\)
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